We present an algorithm for growing the denominator $r$ polygons containing a fixed number of lattice points and enumerate such polygons containing few lattice points for small $r$. We describe …
We present an algorithm for growing the denominator $r$ polygons containing a fixed number of lattice points and enumerate such polygons containing few lattice points for small $r$. We describe the Ehrhart quasi-polynomial of a rational polygon in terms of boundary and interior point counts. Using this, we bound the coefficients of Ehrhart quasi-polynomials of denominator 2 polygons. In particular, we completely classify such polynomials in the case of zero interior points.
We introduce machine learning methodology to the study of lattice polytopes. With supervised learning techniques, we predict standard properties such as volume, dual volume, and reflexivity with accuracies up to …
We introduce machine learning methodology to the study of lattice polytopes. With supervised learning techniques, we predict standard properties such as volume, dual volume, and reflexivity with accuracies up to 100%. We focus on 2d polygons and 3d polytopes with Plücker coordinates as input, which outperform the usual vertex representation.
This is a dataset of randomly generated 8-dimensional Q-factorial Fano toric varieties of Picard rank 2. The data is divided into four plain text files: bound_7_terminal.txt bound_7_non_terminal.txt bound_10_terminal.txt bound_10_non_terminal.txt The …
This is a dataset of randomly generated 8-dimensional Q-factorial Fano toric varieties of Picard rank 2. The data is divided into four plain text files: bound_7_terminal.txt bound_7_non_terminal.txt bound_10_terminal.txt bound_10_non_terminal.txt The numbers 7 and 10 in the file names indicate the bound on the weights used when generating the data. Those varieties with at worst terminal singularities are in the files "bound_N_terminal.txt", and those with non-terminal singularities are in the files "bound_N_non_terminal.txt". The data within each file is de-duplicated, however the data in different files may contain duplicates (for example, it is possible that "bound_7_terminal.txt" and "bound_10_terminal.txt" contain some identical entries). Each line of a file specifies the entries of a (2 x 10)-matrix. For example, the first line of "bound_7_terminal.txt" is: [[5,6,7,7,5,2,5,3,2,2],[0,0,0,1,1,2,6,4,3,3]] and this corresponds to the 8-dimensional Q-factorial Fano toric variety with weight matrix 5 6 7 7 5 2 5 3 2 2 0 0 0 1 1 2 6 4 3 3 and stability condition given by the sum of the columns, which in this case is 44 20 It can be checked that, in this case, the corresponding variety has at worst terminal singularities. In this example the largest occurring weight in the matrix is 7. The number of entries in each file is: bound_7_terminal.txt: 5000000 bound_7_non_terminal.txt: 5000000 bound_10_terminal.txt: 10000000 bound_10_non_terminal.txt: 10000000 For details, see the paper: "Machine learning detects terminal singularities", Tom Coates, Alexander M. Kasprzyk, and Sara Veneziale. Neural Information Processing Systems (NeurIPS), 2023. Magma code capable of generating this dataset is in the file "terminal_dim_8.m". The bound on the weights is set on line 142 by adjusting the value of 'k' (currently set to 10). The target dimension is set on line 143 by adjusting the value of 'dim' (currently set to 8). It is important to note that this code does not attempt to remove duplicates. The code also does not guarantee that the resulting variety has dimension 8. Deduplication and verification of the dimension need to be done separately, after the data has been generated. If you make use of this data, please cite the above paper and the DOI for this data: doi:10.5281/zenodo.10046893
This is a dataset of randomly generated 8-dimensional Q-factorial Fano toric varieties of Picard rank 2. The data is divided into four plain text files: bound_7_terminal.txt bound_7_non_terminal.txt bound_10_terminal.txt bound_10_non_terminal.txt The …
This is a dataset of randomly generated 8-dimensional Q-factorial Fano toric varieties of Picard rank 2. The data is divided into four plain text files: bound_7_terminal.txt bound_7_non_terminal.txt bound_10_terminal.txt bound_10_non_terminal.txt The numbers 7 and 10 in the file names indicate the bound on the weights used when generating the data. Those varieties with at worst terminal singularities are in the files "bound_N_terminal.txt", and those with non-terminal singularities are in the files "bound_N_non_terminal.txt". The data within each file is de-duplicated, however the data in different files may contain duplicates (for example, it is possible that "bound_7_terminal.txt" and "bound_10_terminal.txt" contain some identical entries). Each line of a file specifies the entries of a (2 x 10)-matrix. For example, the first line of "bound_7_terminal.txt" is: [[5,6,7,7,5,2,5,3,2,2],[0,0,0,1,1,2,6,4,3,3]] and this corresponds to the 8-dimensional Q-factorial Fano toric variety with weight matrix 5 6 7 7 5 2 5 3 2 2 0 0 0 1 1 2 6 4 3 3 and stability condition given by the sum of the columns, which in this case is 44 20 It can be checked that, in this case, the corresponding variety has at worst terminal singularities. In this example the largest occurring weight in the matrix is 7. The number of entries in each file is: bound_7_terminal.txt: 5000000 bound_7_non_terminal.txt: 5000000 bound_10_terminal.txt: 10000000 bound_10_non_terminal.txt: 10000000 For details, see the paper: "Machine learning detects terminal singularities", Tom Coates, Alexander M. Kasprzyk, and Sara Veneziale. Neural Information Processing Systems (NeurIPS), 2023. Magma code capable of generating this dataset is in the file "terminal_dim_8.m". The bound on the weights is set on line 142 by adjusting the value of 'k' (currently set to 10). The target dimension is set on line 143 by adjusting the value of 'dim' (currently set to 8). It is important to note that this code does not attempt to remove duplicates. The code also does not guarantee that the resulting variety has dimension 8. Deduplication and verification of the dimension need to be done separately, after the data has been generated. If you make use of this data, please cite the above paper and the DOI for this data: doi:10.5281/zenodo.10046893
Fano varieties are basic building blocks in geometry - they are 'atomic pieces' of mathematical shapes. Recent progress in the classification of Fano varieties involves analysing an invariant called the …
Fano varieties are basic building blocks in geometry - they are 'atomic pieces' of mathematical shapes. Recent progress in the classification of Fano varieties involves analysing an invariant called the quantum period. This is a sequence of integers which gives a numerical fingerprint for a Fano variety. It is conjectured that a Fano variety is uniquely determined by its quantum period. If this is true, one should be able to recover geometric properties of a Fano variety directly from its quantum period. We apply machine learning to the question: does the quantum period of X know the dimension of X? Note that there is as yet no theoretical understanding of this. We show that a simple feed-forward neural network can determine the dimension of X with 98% accuracy. Building on this, we establish rigorous asymptotics for the quantum periods of a class of Fano varieties. These asymptotics determine the dimension of X from its quantum period. Our results demonstrate that machine learning can pick out structure from complex mathematical data in situations where we lack theoretical understanding. They also give positive evidence for the conjecture that the quantum period of a Fano variety determines that variety.
Algebraic varieties are the geometric shapes defined by systems of polynomial equations; they are ubiquitous across mathematics and science. Amongst these algebraic varieties are Q-Fano varieties: positively curved shapes which …
Algebraic varieties are the geometric shapes defined by systems of polynomial equations; they are ubiquitous across mathematics and science. Amongst these algebraic varieties are Q-Fano varieties: positively curved shapes which have Q-factorial terminal singularities. Q-Fano varieties are of fundamental importance in geometry as they are "atomic pieces" of more complex shapes - the process of breaking a shape into simpler pieces in this sense is called the Minimal Model Programme. Despite their importance, the classification of Q-Fano varieties remains unknown. In this paper we demonstrate that machine learning can be used to understand this classification. We focus on 8-dimensional positively-curved algebraic varieties that have toric symmetry and Picard rank 2, and develop a neural network classifier that predicts with 95% accuracy whether or not such an algebraic variety is Q-Fano. We use this to give a first sketch of the landscape of Q-Fanos in dimension 8. How the neural network is able to detect Q-Fano varieties with such accuracy remains mysterious, and hints at some deep mathematical theory waiting to be uncovered. Furthermore, when visualised using the quantum period, an invariant that has played an important role in recent theoretical developments, we observe that the classification as revealed by ML appears to fall within a bounded region, and is stratified by the Fano index. This suggests that it may be possible to state and prove conjectures on completeness in the future. Inspired by the ML analysis, we formulate and prove a new global combinatorial criterion for a positively curved toric variety of Picard rank 2 to have terminal singularities. Together with the first sketch of the landscape of Q-Fanos in higher dimensions, this gives new evidence that machine learning can be an essential tool in developing mathematical conjectures and accelerating theoretical discovery.
This dataset contains certain rigid maximally mutable Laurent polynomials (rigid MMLPs) in three variables. Rigid MMLPs are defined in reference [1]. The Newton polytopes of these Laurent polynomials are three-dimensional …
This dataset contains certain rigid maximally mutable Laurent polynomials (rigid MMLPs) in three variables. Rigid MMLPs are defined in reference [1]. The Newton polytopes of these Laurent polynomials are three-dimensional canonical Fano polytopes. That is, they are three-dimensional convex polytopes with vertices that are primitive integer vectors and that contain exactly one lattice point, the origin, in their strict interior. See references [2] and [3]. Although the rigid MMLPs specified in this dataset have 3-dimensional canonical Fano Newton polytope, this is by no means an exhaustive list of such Laurent polynomials. The dataset contains examples of rigid MMLPs that correspond under mirror symmetry to three-dimensional Q-Fano varieties of particulaly high estimated codimension: see reference [4]. The file "rigid_MMLPs.txt" contains key:value records with keys and values as described below, separated by blank lines. Each key:value record determines a rigid MMLP, and there are 130 records in the file. An example record is: canonical3_id: 231730<br> coefficients: [1,1,1,1,1,1,1,1,1,1]<br> exponents: [[-1,-1,-1],[0,1,0],[0,1,1],[1,0,0],[1,0,1],[1,2,2],[2,1,2],[2,1,3],[3,3,5],[4,2,5]]<br> period: [1,0,0,12,24,0,540,2940,2520,33600,327600,693000,2795100,35315280,129909780,354666312,3816572760,20559258720,59957561664,435508321248,2969362219824]<br> ulid: 01G5CBH3F86NRYF0TJ8MYWM41H The keys and values are as follows, where f denotes the Laurent polynomial defined by the key:value record. canonical3_id: an integer, the ID of the Newton polytope of f in reference [3]<br> coefficients: a string of the form "[c1,c2,...,cN]" where c1, c2, ... are integers. These are the coefficients of f.<br> exponents: a string of the form "[[x1,y1,z1],[x2,y2,z2],...,[xN,yN,zN]]" where x1, y1, z1, ..., xN, yN, zN are integers. These are the exponents of f.<br> period: a string of the form "[d0,d1,...,d20]" where d0, d1, ..., d20 are non-negative integers that give the first 21 terms of the period sequence for f.<br> ulid: a string that uniquely identified this entry in the dataset The sequences defined by the keys "coefficients" and "exponents" are parallel to each other. The period sequence for f is defined, for example, in equations 1.2 and 1.3 of reference [1]. References [1] Tom Coates, Alexander M. Kasprzyk, Giuseppe Pitton, and Ketil Tveiten. Maximally mutable Laurent polynomials. Proceedings of the Royal Society A 477, no. 2254:20210584, 2021. [2] Alexander M. Kasprzyk. Canonical toric Fano threefolds. Canadian Journal of Mathematics, 62(6):1293–1309, 2010. [3] Alexander M. Kasprzyk. The classification of toric canonical Fano 3-folds. Zenodo, https://doi.org/10.5281/zenodo.5866330, 2010. [4] Liana Heuberger. Q-Fano threefolds and Laurent inversion. Preprint, arXiv:2202.04184, 2022.
This dataset contains certain rigid maximally mutable Laurent polynomials (rigid MMLPs) in three variables. Rigid MMLPs are defined in reference [1]. The Newton polytopes of these Laurent polynomials are three-dimensional …
This dataset contains certain rigid maximally mutable Laurent polynomials (rigid MMLPs) in three variables. Rigid MMLPs are defined in reference [1]. The Newton polytopes of these Laurent polynomials are three-dimensional canonical Fano polytopes. That is, they are three-dimensional convex polytopes with vertices that are primitive integer vectors and that contain exactly one lattice point, the origin, in their strict interior. See references [2] and [3]. Although the rigid MMLPs specified in this dataset have 3-dimensional canonical Fano Newton polytope, this is by no means an exhaustive list of such Laurent polynomials. The dataset contains examples of rigid MMLPs that correspond under mirror symmetry to three-dimensional Q-Fano varieties of particulaly high estimated codimension: see reference [4]. The file "rigid_MMLPs.txt" contains key:value records with keys and values as described below, separated by blank lines. Each key:value record determines a rigid MMLP, and there are 130 records in the file. An example record is: canonical3_id: 231730<br> coefficients: [1,1,1,1,1,1,1,1,1,1]<br> exponents: [[-1,-1,-1],[0,1,0],[0,1,1],[1,0,0],[1,0,1],[1,2,2],[2,1,2],[2,1,3],[3,3,5],[4,2,5]]<br> period: [1,0,0,12,24,0,540,2940,2520,33600,327600,693000,2795100,35315280,129909780,354666312,3816572760,20559258720,59957561664,435508321248,2969362219824]<br> ulid: 01G5CBH3F86NRYF0TJ8MYWM41H The keys and values are as follows, where f denotes the Laurent polynomial defined by the key:value record. canonical3_id: an integer, the ID of the Newton polytope of f in reference [3]<br> coefficients: a string of the form "[c1,c2,...,cN]" where c1, c2, ... are integers. These are the coefficients of f.<br> exponents: a string of the form "[[x1,y1,z1],[x2,y2,z2],...,[xN,yN,zN]]" where x1, y1, z1, ..., xN, yN, zN are integers. These are the exponents of f.<br> period: a string of the form "[d0,d1,...,d20]" where d0, d1, ..., d20 are non-negative integers that give the first 21 terms of the period sequence for f.<br> ulid: a string that uniquely identified this entry in the dataset The sequences defined by the keys "coefficients" and "exponents" are parallel to each other. The period sequence for f is defined, for example, in equations 1.2 and 1.3 of reference [1]. References [1] Tom Coates, Alexander M. Kasprzyk, Giuseppe Pitton, and Ketil Tveiten. Maximally mutable Laurent polynomials. Proceedings of the Royal Society A 477, no. 2254:20210584, 2021. [2] Alexander M. Kasprzyk. Canonical toric Fano threefolds. Canadian Journal of Mathematics, 62(6):1293–1309, 2010. [3] Alexander M. Kasprzyk. The classification of toric canonical Fano 3-folds. Zenodo, https://doi.org/10.5281/zenodo.5866330, 2010. [4] Liana Heuberger. Q-Fano threefolds and Laurent inversion. Preprint, arXiv:2202.04184, 2022.
<strong>Ehrhart series coefficients and quasi-period for random rational polytopes</strong> A dataset of Ehrhart data for 84000 randomly generated rational polytopes, in dimensions 2 to 4, with quasi-periods 2 to 15. …
<strong>Ehrhart series coefficients and quasi-period for random rational polytopes</strong> A dataset of Ehrhart data for 84000 randomly generated rational polytopes, in dimensions 2 to 4, with quasi-periods 2 to 15. The polytopes used to generate this data were produced by the following algorithm: Fix \(d\) a positive integer in \(\{2,3,4\}\). Choose \(r\in\{2,\ldots,15\}\) uniformly at random. Choose \(d + k\) lattice points \(\{v_1,\ldots,v_{d+k}\}\) uniformly at random in a box \([-5r,5r]^d\), where \(k\) is chosen uniformly at random in \(\{1,\ldots,5\}\). Set \(P := \mathrm{conv}\{v_1,\ldots,v_{d+k}\}\). If \(\mathrm{dim}(P)\) is not equal to \(d\) then return to step 3. Choose a lattice point \(v\in P \cap \mathbb{Z}^d\) uniformly at random and replace \(P\) with the translation \(P-v\). Replace \(P\) with the dilation \(P/r\). The final dataset was produced by first removing duplicate records, and then downsampling to a subset with 2000 datapoints for each pair \((d,q)\), where \(d\) is the dimension of \(P\) and \(q\) is the quasi-period of \(P\), with \(d\in\{2,3,4\}\) and \(q\in\{2,\ldots,15\}\). For details, see the paper: <em>Machine Learning the Dimension of a Polytope</em>, Tom Coates, Johannes Hofscheier, and Alexander M. Kasprzyk, 2022. If you make use of this data, please cite the above paper and the DOI for this data: doi:10.5281/zenodo.6614829 <strong>quasiperiod.txt.gz</strong><br> The file "quasiperiod.txt.gz" is a gzip-compressed plain text file containing key:value records with keys and values as described below, where each record is separated by a blank line. There are 84000 records in the file. <strong>Example record</strong><br> ULID: 01G57JBYP2ZW825E0NT4Q9JQNQ<br> Dimension: 2<br> Quasiperiod: 2<br> Volume: 97<br> EhrhartDelta: [1,50,195,289,192,49]<br> Ehrhart: [1,50,198,...]<br> LogEhrhart: [0.000000000000000000000000000000,3.91202300542814605861875078791,5.28826703069453523626966617327,...] (The values for Ehrhart and LogEhrhart in the example have been truncated.) For each polytope \(P\) of dimension \(d\) and quasi-period \(q\) we record the following keys and values in the dataset: ULID: A randomly generated string identifier for this record.<br> Dimension: A positive integer. The dimension \(2 \leq d \leq 4\) of the polytope \(P\).<br> Quasiperiod: A positive integer. The quasi-period \(2 \leq q \leq 15\) of the polytope \(P\).<br> Volume: A positive rational number. The lattice-normalised volume \(\mathrm{Vol}(P)\) of the polytope \(P\).<br> EhrhartDelta: A sequence \([1,a_1,a_2,\ldots,a_N]\) of integers of length \(N + 1\), where \(N := q(d + 1) - 1\). This is the Ehrhart \(\delta\)-vector (or \(h^*\)-vector) of \(P\). The Ehrhart series \(\mathrm{Ehr}(P)\) of \(P\) is given by the power-series expansion of \((1 + a_1t + a_2t^2 + \ldots + a_Nt^N) / (1 - t^q)^{d+1}\).<br> Ehrhart: A sequence \([1,c_1,c_2,\ldots,c_{1100}]\) of positive integers. The value \(c_i\) is equal to the number of lattice points in the \(i\)-th dilation of \(P\), that is, \(c_i = \#(iP \cap \mathbb{Z}^d)\). Equivalently, \(c_i\) is the coefficient of \(t^i\) in \(\mathrm{Ehr}(P) = 1 + c_1t + c_2t^2 + \ldots = (1 + a_1t + a_2t^2 + \ldots + a_Nt^N) / (1 - t^q)^{d+1}\).<br> LogEhrhart: A sequence \([0,y_1,y_2,\ldots,y_{1100}]\) of non-negative floating point numbers. Here \(y_i := \log c_i\).
<strong>Ehrhart series coefficients for random lattice polytopes</strong> A dataset of Ehrhart data for 2918 randomly generated lattice polytopes, in dimensions 2 to 8. The polytopes used to generate this data …
<strong>Ehrhart series coefficients for random lattice polytopes</strong> A dataset of Ehrhart data for 2918 randomly generated lattice polytopes, in dimensions 2 to 8. The polytopes used to generate this data were produced by the following algorithm: Fix \(d\) a positive integer in \(\{2,\ldots,8\}\). Choose \(d + k\) lattice points \(\{v_1,\ldots,v_{d+k}\}\) uniformly at random in a box \([-5,5]^d\), where \(k\) is chosen uniformly at random in \(\{1,\ldots,5\}\). Set \(P := \mathrm{conv}\{v_1,\ldots,v_{d+k}\}\). If \(\mathrm{dim}(P)\neq d\) then return to step 2. The final dataset has duplicate records removed. The data is distributed by dimension \(d\) as follows: d 2 3 4 5 6 7 8 # 431 787 812 399 181 195 113 For details, see the paper: <em>Machine Learning the Dimension of a Polytope</em>, Tom Coates, Johannes Hofscheier, and Alexander M. Kasprzyk, 2022. If you make use of this data, please cite the above paper and the DOI for this data: doi:10.5281/zenodo.6614821 <strong>dimension.txt.gz</strong><br> The file "dimension.txt.gz" is a gzip-compressed plain text file containing key:value records with keys and values as described below, where each record is separated by a blank line. There are 2918 records in the file. <strong>Example record</strong><br> ULID: 1FTU9VGPXXU82CTDGD6WYMBF9<br> Dimension: 3<br> Volume: 342<br> EhrhartDelta: [1,70,223,48]<br> Ehrhart: [1,74,513,...]<br> LogEhrhart: [0.000000000000000000000000000000,4.30406509320416975378532779249,6.24027584517076953419476314266,...] (The values for Ehrhart and LogEhrhart in the example have been truncated.) For each polytope \(P\) of dimension \(d\) we record the following keys and values in the dataset: ULID: A randomly generated string identifier for this record.<br> Dimension: A positive integer. The dimension \(2 \leq d \leq 8\) of the polytope \(P\).<br> Volume: A positive integer. The lattice-normalised volume \(\mathrm{Vol}(P)\) of the polytope \(P\).<br> EhrhartDelta: A sequence \([1,a_1,a_2,\ldots,a_d]\) of integers of length \(d + 1\). This is the Ehrhart \(\delta\)-vector (or \(h^*\)-vector) of \(P\). The Ehrhart series \(\mathrm{Ehr}(P)\) of \(P\) is given by the power-series expansion of \((1 + a_1t + a_2t^2 + \ldots + a_dt^d) / (1 - t)^{d+1}\). In particular, \(\mathrm{Vol}(P) = 1 + a_1 + a_2 + \ldots + a_d\).<br> Ehrhart: A sequence \([1,c_1,c_2,\ldots,c_{1100}]\) of positive integers. The value \(c_i\) is equal to the number of lattice points in the \(i\)-th dilation of \(P\), that is, \(c_i = \#(iP \cap \mathbb{Z}^d)\). Equivalently, \(c_i\) is the coefficient of \(t^i\) in \(\mathrm{Ehr}(P) = 1 + c_1t + c_2t^2 + \ldots = (1 + a_1t + a_2t^2 + \ldots + a_dt^d) / (1 - t)^{d+1}\).<br> LogEhrhart: A sequence \([0,y_1,y_2,\ldots,y_{1100}]\) of non-negative floating point numbers. Here \(y_i := \log c_i\)
<strong>Ehrhart series coefficients for random lattice polytopes</strong> A dataset of Ehrhart data for 2918 randomly generated lattice polytopes, in dimensions 2 to 8. The polytopes used to generate this data …
<strong>Ehrhart series coefficients for random lattice polytopes</strong> A dataset of Ehrhart data for 2918 randomly generated lattice polytopes, in dimensions 2 to 8. The polytopes used to generate this data were produced by the following algorithm: Fix \(d\) a positive integer in \(\{2,\ldots,8\}\). Choose \(d + k\) lattice points \(\{v_1,\ldots,v_{d+k}\}\) uniformly at random in a box \([-5,5]^d\), where \(k\) is chosen uniformly at random in \(\{1,\ldots,5\}\). Set \(P := \mathrm{conv}\{v_1,\ldots,v_{d+k}\}\). If \(\mathrm{dim}(P)\neq d\) then return to step 2. The final dataset has duplicate records removed. The data is distributed by dimension \(d\) as follows: d 2 3 4 5 6 7 8 # 431 787 812 399 181 195 113 For details, see the paper: <em>Machine Learning the Dimension of a Polytope</em>, Tom Coates, Johannes Hofscheier, and Alexander M. Kasprzyk, 2022. If you make use of this data, please cite the above paper and the DOI for this data: doi:10.5281/zenodo.6614821 <strong>dimension.txt.gz</strong><br> The file "dimension.txt.gz" is a gzip-compressed plain text file containing key:value records with keys and values as described below, where each record is separated by a blank line. There are 2918 records in the file. <strong>Example record</strong><br> ULID: 1FTU9VGPXXU82CTDGD6WYMBF9<br> Dimension: 3<br> Volume: 342<br> EhrhartDelta: [1,70,223,48]<br> Ehrhart: [1,74,513,...]<br> LogEhrhart: [0.000000000000000000000000000000,4.30406509320416975378532779249,6.24027584517076953419476314266,...] (The values for Ehrhart and LogEhrhart in the example have been truncated.) For each polytope \(P\) of dimension \(d\) we record the following keys and values in the dataset: ULID: A randomly generated string identifier for this record.<br> Dimension: A positive integer. The dimension \(2 \leq d \leq 8\) of the polytope \(P\).<br> Volume: A positive integer. The lattice-normalised volume \(\mathrm{Vol}(P)\) of the polytope \(P\).<br> EhrhartDelta: A sequence \([1,a_1,a_2,\ldots,a_d]\) of integers of length \(d + 1\). This is the Ehrhart \(\delta\)-vector (or \(h^*\)-vector) of \(P\). The Ehrhart series \(\mathrm{Ehr}(P)\) of \(P\) is given by the power-series expansion of \((1 + a_1t + a_2t^2 + \ldots + a_dt^d) / (1 - t)^{d+1}\). In particular, \(\mathrm{Vol}(P) = 1 + a_1 + a_2 + \ldots + a_d\).<br> Ehrhart: A sequence \([1,c_1,c_2,\ldots,c_{1100}]\) of positive integers. The value \(c_i\) is equal to the number of lattice points in the \(i\)-th dilation of \(P\), that is, \(c_i = \#(iP \cap \mathbb{Z}^d)\). Equivalently, \(c_i\) is the coefficient of \(t^i\) in \(\mathrm{Ehr}(P) = 1 + c_1t + c_2t^2 + \ldots = (1 + a_1t + a_2t^2 + \ldots + a_dt^d) / (1 - t)^{d+1}\).<br> LogEhrhart: A sequence \([0,y_1,y_2,\ldots,y_{1100}]\) of non-negative floating point numbers. Here \(y_i := \log c_i\)
<strong>Ehrhart series coefficients and quasi-period for random rational polytopes</strong> A dataset of Ehrhart data for 84000 randomly generated rational polytopes, in dimensions 2 to 4, with quasi-periods 2 to 15. …
<strong>Ehrhart series coefficients and quasi-period for random rational polytopes</strong> A dataset of Ehrhart data for 84000 randomly generated rational polytopes, in dimensions 2 to 4, with quasi-periods 2 to 15. The polytopes used to generate this data were produced by the following algorithm: Fix \(d\) a positive integer in \(\{2,3,4\}\). Choose \(r\in\{2,\ldots,15\}\) uniformly at random. Choose \(d + k\) lattice points \(\{v_1,\ldots,v_{d+k}\}\) uniformly at random in a box \([-5r,5r]^d\), where \(k\) is chosen uniformly at random in \(\{1,\ldots,5\}\). Set \(P := \mathrm{conv}\{v_1,\ldots,v_{d+k}\}\). If \(\mathrm{dim}(P)\) is not equal to \(d\) then return to step 3. Choose a lattice point \(v\in P \cap \mathbb{Z}^d\) uniformly at random and replace \(P\) with the translation \(P-v\). Replace \(P\) with the dilation \(P/r\). The final dataset was produced by first removing duplicate records, and then downsampling to a subset with 2000 datapoints for each pair \((d,q)\), where \(d\) is the dimension of \(P\) and \(q\) is the quasi-period of \(P\), with \(d\in\{2,3,4\}\) and \(q\in\{2,\ldots,15\}\). For details, see the paper: <em>Machine Learning the Dimension of a Polytope</em>, Tom Coates, Johannes Hofscheier, and Alexander M. Kasprzyk, 2022. If you make use of this data, please cite the above paper and the DOI for this data: doi:10.5281/zenodo.6614829 <strong>quasiperiod.txt.gz</strong><br> The file "quasiperiod.txt.gz" is a gzip-compressed plain text file containing key:value records with keys and values as described below, where each record is separated by a blank line. There are 84000 records in the file. <strong>Example record</strong><br> ULID: 01G57JBYP2ZW825E0NT4Q9JQNQ<br> Dimension: 2<br> Quasiperiod: 2<br> Volume: 97<br> EhrhartDelta: [1,50,195,289,192,49]<br> Ehrhart: [1,50,198,...]<br> LogEhrhart: [0.000000000000000000000000000000,3.91202300542814605861875078791,5.28826703069453523626966617327,...] (The values for Ehrhart and LogEhrhart in the example have been truncated.) For each polytope \(P\) of dimension \(d\) and quasi-period \(q\) we record the following keys and values in the dataset: ULID: A randomly generated string identifier for this record.<br> Dimension: A positive integer. The dimension \(2 \leq d \leq 4\) of the polytope \(P\).<br> Quasiperiod: A positive integer. The quasi-period \(2 \leq q \leq 15\) of the polytope \(P\).<br> Volume: A positive rational number. The lattice-normalised volume \(\mathrm{Vol}(P)\) of the polytope \(P\).<br> EhrhartDelta: A sequence \([1,a_1,a_2,\ldots,a_N]\) of integers of length \(N + 1\), where \(N := q(d + 1) - 1\). This is the Ehrhart \(\delta\)-vector (or \(h^*\)-vector) of \(P\). The Ehrhart series \(\mathrm{Ehr}(P)\) of \(P\) is given by the power-series expansion of \((1 + a_1t + a_2t^2 + \ldots + a_Nt^N) / (1 - t^q)^{d+1}\).<br> Ehrhart: A sequence \([1,c_1,c_2,\ldots,c_{1100}]\) of positive integers. The value \(c_i\) is equal to the number of lattice points in the \(i\)-th dilation of \(P\), that is, \(c_i = \#(iP \cap \mathbb{Z}^d)\). Equivalently, \(c_i\) is the coefficient of \(t^i\) in \(\mathrm{Ehr}(P) = 1 + c_1t + c_2t^2 + \ldots = (1 + a_1t + a_2t^2 + \ldots + a_Nt^N) / (1 - t^q)^{d+1}\).<br> LogEhrhart: A sequence \([0,y_1,y_2,\ldots,y_{1100}]\) of non-negative floating point numbers. Here \(y_i := \log c_i\).
Fano manifolds are basic building blocks in geometry - they are, in a precise sense, atomic pieces of shapes. The classification of Fano manifolds is therefore an important problem in …
Fano manifolds are basic building blocks in geometry - they are, in a precise sense, atomic pieces of shapes. The classification of Fano manifolds is therefore an important problem in geometry, which has been open since the 1930s. One can think of this as building a Periodic Table for shapes. A recent breakthrough in Fano classification involves a technique from theoretical physics called Mirror Symmetry. From this perspective, a Fano manifold is encoded by a sequence of integers: the coefficients of a power series called the regularized quantum period. Progress to date has been hindered by the fact that quantum periods require specialist expertise to compute, and descriptions of known Fano manifolds and their regularized quantum periods are incomplete and scattered in the literature. We describe databases of regularized quantum periods for Fano manifolds in dimensions up to four. The databases in dimensions one, two, and three are complete; the database in dimension four will be updated as new four-dimensional Fano manifolds are discovered and new regularized quantum periods computed.
We survey the approach to mirror symmetry via Laurent polynomials, outlining some of the main conjectures, problems, and questions related to the subject. We discuss: how to construct Landau--Ginzburg models …
We survey the approach to mirror symmetry via Laurent polynomials, outlining some of the main conjectures, problems, and questions related to the subject. We discuss: how to construct Landau--Ginzburg models for Fano varieties; how to apply them to classification problems; and how to compute invariants of Fano varieties via Landau--Ginzburg models.
We describe how simple machine learning methods successfully predict geometric properties from Hilbert series (HS). Regressors predict embedding weights in projective space to ${\sim}1$ mean absolute error, whilst classifiers predict …
We describe how simple machine learning methods successfully predict geometric properties from Hilbert series (HS). Regressors predict embedding weights in projective space to ${\sim}1$ mean absolute error, whilst classifiers predict dimension and Gorenstein index to $>90\%$ accuracy with ${\sim}0.5\%$ standard error. Binary random forest classifiers managed to distinguish whether the underlying HS describes a complete intersection with high accuracies exceeding $95\%$. Neural networks (NNs) exhibited success identifying HS from a Gorenstein ring to the same order of accuracy, whilst generation of 'fake' HS proved trivial for NNs to distinguish from those associated to the three-dimensional Fano varieties considered.
<strong>Toric varieties of Picard rank 2 with at worst terminal singularities</strong> A dataset of 200000 randomly generated toric varieties of Picard rank 2 with at worst terminal Q-factorial singularities, in …
<strong>Toric varieties of Picard rank 2 with at worst terminal singularities</strong> A dataset of 200000 randomly generated toric varieties of Picard rank 2 with at worst terminal Q-factorial singularities, in dimensions 2 to 10. The data consists of the plain text files "rank_2_dim_N.txt" where N, which is the dimension of the toric variety, is in the range 2 to 10. Each line of the file specifies the entries of a (2 x N+2)-matrix. For example, the first line of "rank_2_dim_4.txt" is: [[1,3,5,4,1,0],[0,1,2,5,3,1]] and this corresponds to the 4-dimensional toric variety with weight matrix 1 3 5 4 1 0<br> 0 1 2 5 3 1 and stability condition given by the sum of the columns, which in this case is 14<br> 12 For details, see the paper: "Machine learning the dimension of a Fano variety", Tom Coates, Alexander M. Kasprzyk, and Sara Veneziale, <em>Nature Communications</em>, <strong>14:</strong>5526 (2023). doi:10.1038/s41467-023-41157-1 Magma code capable of generating this dataset is in the file "generate_rank_2.m". If you make use of this data, please cite the above paper and the DOI for this data: doi:10.5281/zenodo.5790096
<strong>Toric varieties of Picard rank 2 with at worst terminal singularities</strong> A dataset of 200000 randomly generated toric varieties of Picard rank 2 with at worst terminal Q-factorial singularities, in …
<strong>Toric varieties of Picard rank 2 with at worst terminal singularities</strong> A dataset of 200000 randomly generated toric varieties of Picard rank 2 with at worst terminal Q-factorial singularities, in dimensions 2 to 10. The data consists of the plain text files "rank_2_dim_N.txt" where N, which is the dimension of the toric variety, is in the range 2 to 10. Each line of the file specifies the entries of a (2 x N+2)-matrix. For example, the first line of "rank_2_dim_4.txt" is: [[1,3,5,4,1,0],[0,1,2,5,3,1]] and this corresponds to the 4-dimensional toric variety with weight matrix 1 3 5 4 1 0<br> 0 1 2 5 3 1 and stability condition given by the sum of the columns, which in this case is 14<br> 12 For details, see the paper: "Machine learning the dimension of a Fano variety", Tom Coates, Alexander M. Kasprzyk, and Sara Veneziale, <em>Nature Communications</em>, <strong>14:</strong>5526 (2023). doi:10.1038/s41467-023-41157-1 Magma code capable of generating this dataset is in the file "generate_rank_2.m". If you make use of this data, please cite the above paper and the DOI for this data: doi:10.5281/zenodo.5790096
Abstract We give an upper bound on the volume $\operatorname {vol}(P^*)$ of a polytope $P^*$ dual to a d -dimensional lattice polytope P with exactly one interior lattice point in …
Abstract We give an upper bound on the volume $\operatorname {vol}(P^*)$ of a polytope $P^*$ dual to a d -dimensional lattice polytope P with exactly one interior lattice point in each dimension d . This bound, expressed in terms of the Sylvester sequence, is sharp and achieved by the dual to a particular reflexive simplex. Our result implies a sharp upper bound on the volume of a d -dimensional reflexive polytope. Translated into toric geometry, this gives a sharp upper bound on the anti-canonical degree $(-K_X)^d$ of a d -dimensional Fano toric variety X with at worst canonical singularities.
We explain an effective Kawamata boundedness result for Mori-Fano 3-folds. In particular, we describe a list of 39,550 possible Hilbert series of semistable Mori-Fano 3-folds, with examples to explain its …
We explain an effective Kawamata boundedness result for Mori-Fano 3-folds. In particular, we describe a list of 39,550 possible Hilbert series of semistable Mori-Fano 3-folds, with examples to explain its meaning, its relationship to known classifications and the wealth of more general Fano 3-folds it contains, as well as its application to the on-going classification of Fano 3-folds.
We explain a web of Sarkisov links that overlies the classification of Fano weighted projective spaces in dimensions 3 and 4, extending results of Prokhorov.
We explain a web of Sarkisov links that overlies the classification of Fano weighted projective spaces in dimensions 3 and 4, extending results of Prokhorov.
We use machine learning to predict the dimension of a lattice polytope directly from its Ehrhart series. This is highly effective, achieving almost 100% accuracy. We also use machine learning …
We use machine learning to predict the dimension of a lattice polytope directly from its Ehrhart series. This is highly effective, achieving almost 100% accuracy. We also use machine learning to recover the volume of a lattice polytope from its Ehrhart series, and to recover the dimension, volume, and quasi-period of a rational polytope from its Ehrhart series. In each case we achieve very high accuracy, and we propose mathematical explanations for why this should be so.
We describe recent progress in a program to understand the classification of three-dimensional Fano varieties with $\mathbb{Q}$-factorial terminal singularities using mirror symmetry. As part of this we give an improved …
We describe recent progress in a program to understand the classification of three-dimensional Fano varieties with $\mathbb{Q}$-factorial terminal singularities using mirror symmetry. As part of this we give an improved and more conceptual understanding of Laurent inversion, a technique that sometimes allows one to construct a Fano variety $X$ directly from a Laurent polynomial $f$ that corresponds to it under mirror symmetry.
The family of smooth Fano 3-folds with Picard rank 1 and anticanonical volume 4 consists of quartic 3-folds and of double covers of the 3-dimensional quadric branched along an octic …
The family of smooth Fano 3-folds with Picard rank 1 and anticanonical volume 4 consists of quartic 3-folds and of double covers of the 3-dimensional quadric branched along an octic surface. They can all be parametrised as complete intersections of a quadric and a quartic in the weighted projective space $\mathbb{P}(1,1,1,1,1,2)$, denoted by $X_{2,4} \subset \mathbb{P}(1^5,2)$; all such smooth complete intersections are K-stable. With the aim of investigating the compactification of the moduli space of quartic 3-folds given by K-stability, we exhibit three phenomena: (i) there exist K-polystable complete intersection $X_{2,2,4} \subset \mathbb{P}(1^5,2^2)$ Fano 3-folds which deform to quartic 3-folds and are neither quartic 3-folds nor double covers of quadric 3-folds - in other words, the closure of the locus parametrising complete intersections $X_{2,4}\subset \mathbb{P}(1^5,2)$ in the K-moduli contains elements that are not of this type; (ii) any quasi-smooth $X_{2,2,4} \subset \mathbb{P}(1^5,2^2)$ is K-polystable; (iii) the closure in the K-moduli space of the locus parametrising complete intersections $X_{2,2,4} \subset \mathbb{P}(1^5,2^2)$ which are not complete intersections $X_{2,4} \subset \mathbb{P}(1^5,2)$ contains only points which correspond to complete intersections $X_{2,2,4} \subset \mathbb{P}(1^5,2^2)$.
Fano varieties are 'atomic pieces' of algebraic varieties, the shapes that can be defined by polynomial equations. We describe the role of computation and database methods in the construction and …
Fano varieties are 'atomic pieces' of algebraic varieties, the shapes that can be defined by polynomial equations. We describe the role of computation and database methods in the construction and classification of Fano varieties, with an emphasis on three-dimensional Fano varieties with mild singularities called Q-Fano threefolds. The classification of Q-Fano threefolds has been open for several decades, but there has been significant recent progress. These advances combine computational algebraic geometry and large-scale data analysis with new ideas that originated in theoretical physics.
We survey the approach to mirror symmetry via Laurent polynomials, outlining some of the main conjectures, problems, and questions related to the subject. We discuss: how to construct Landau--Ginzburg models …
We survey the approach to mirror symmetry via Laurent polynomials, outlining some of the main conjectures, problems, and questions related to the subject. We discuss: how to construct Landau--Ginzburg models for Fano varieties; how to apply them to classification problems; and how to compute invariants of Fano varieties via Landau--Ginzburg models.
<strong>The database smooth_fano_2</strong> This is a database of regularized quantum periods for two-dimensional Fano manifolds. There are ten entries in the database. Each entry in the database is a key-value …
<strong>The database smooth_fano_2</strong> This is a database of regularized quantum periods for two-dimensional Fano manifolds. There are ten entries in the database. Each entry in the database is a key-value record with keys and values as described in the paper: <em>Databases of Quantum Periods for Fano Manifolds</em>, Tom Coates and Alexander M. Kasprzyk, 2021. If you make use of this data, please cite the above paper and the DOI for this data: doi:10.5281/zenodo.5708232
<strong>The database smooth_fano_3</strong> This is a database of regularized quantum periods for three-dimensional Fano manifolds. There are 105 entries in the database. Each entry in the database is a key-value …
<strong>The database smooth_fano_3</strong> This is a database of regularized quantum periods for three-dimensional Fano manifolds. There are 105 entries in the database. Each entry in the database is a key-value record with keys and values as described in the paper: <em>Databases of Quantum Periods for Fano Manifolds</em>, Tom Coates and Alexander M. Kasprzyk, 2021. If you make use of this data, please cite the above paper and the DOI for this data: doi:10.5281/zenodo.5708272
<strong>The database smooth_fano_4</strong> This is a database of regularized quantum periods for four-dimensional Fano manifolds. The database will be updated as new four-dimensional Fano manifolds are discovered and new regularized …
<strong>The database smooth_fano_4</strong> This is a database of regularized quantum periods for four-dimensional Fano manifolds. The database will be updated as new four-dimensional Fano manifolds are discovered and new regularized quantum periods computed. Each entry in the database is a key-value record with keys and values as described in the paper [CK2021]. If you make use of this data, please cite that paper and the DOI for this data: doi:10.5281/zenodo.5708307 <strong>Names</strong> The database describes Fano varieties via names, as follows: Names of Fano manifolds Name Description P1 one-dimensional projective space P2 two-dimensional projective space dP(k) the del Pezzo surface of degree k given by the blow-up of P2 in 9-k points P3 three-dimensional projective space Q3 a quadric hypersurface in four-dimensional projective space B(3,k) the three-dimensional Fano manifold of Picard rank 1, Fano index 2, and degree 8k V(3,k) the three-dimensional Fano manifold of Picard rank 1, Fano index 1, and degree k MM(r,k) the k-th entry in the Mori-Mukai list of three-dimensional Fano manifolds of Picard rank r, ordered as in [CCGK2016]<br> P4 four-dimensional projective space Q4 a quadric hypersurface in five-dimensional projective space FI(4,k) the four-dimensional Fano manifold of Fano index 3 and degree 81k V(4,k) the four-dimensional Fano manifold of Picard rank 1, Fano index 2, and degree 16k MW(4,k) the k-th entry in Table 12.7 of [IP1999] of four-dimensional Fano manifolds of Fano index 2 and Picard rank greater than 1 Obro(4,k) the k-th four-dimensional Fano toric manifold in Obro's classification [O2007] Str(k) the k-th Strangeway manifold in [CGKS2020] CKP(k) the k-th four-dimensional Fano toric complete intersection in [CKP2015] CKK(k) the k-th four-dimensional Fano quiver flag zero locus in Appendix B of [K2019] A name of the form "S1 x S2", where S1 and S2 are names of Fano manifolds X1 and X2, refers to the product manifold X1 x X2. <strong>References</strong> [CCGK2016] <em>Quantum periods for 3-dimensional Fano manifolds</em>; Tom Coates, Alessio Corti, Sergey Galkin, Alexander M. Kasprzyk; Geometry and Topology 20 (2016), no. 1, 103-256. [CGKS2020] <em>Quantum periods for certain four-dimensional Fano manifolds</em>; Tom Coates, Sergey Galkin, Alexander M. Kasprzyk, Andrew Strangeway; Experimental Math. 29 (2020), no. 2, 183-221. [CK2021] <em>Databases of quantum periods for Fano manifolds</em>; Tom Coates, Alexander M. Kasprzyk; 2021. [CKP2015] <em>Four-dimensional Fano toric complete intersections</em>; Tom Coates, Alexander M. Kasprzyk, Thomas Prince; Proc. Royal Society A 471 (2015), no. 2175, 20140704, 14. [IP1999] <em>Fano varieties</em>; V.A. Iskovskikh, Yu. G. Prokhorov; Encyclopaedia Math. Sci. vol. 47, Springer, Berlin, 1999, 1-247. [K2019] <em>Four-dimensional Fano quiver flag zero loci</em>; Elana Kalashnikov; Proc. Royal Society A 275 (2019), no. 2225, 20180791, 23. [O2007] <em>An algorithm for the classification of smooth Fano polytopes</em>; Mikkel Obro; arXiv:0704.0049 [math.CO]; 2007.
<strong>The database smooth_fano_1</strong> This is a database of regularized quantum periods for one-dimensional Fano manifolds. There is one entry in the database. Each entry in the database is a key-value …
<strong>The database smooth_fano_1</strong> This is a database of regularized quantum periods for one-dimensional Fano manifolds. There is one entry in the database. Each entry in the database is a key-value record with keys and values as described in the paper: <em>Databases of Quantum Periods for Fano Manifolds</em>, Tom Coates and Alexander M. Kasprzyk, 2021. If you make use of this data, please cite the above paper and the DOI for this data: doi:10.5281/zenodo.5708188
<strong>The database smooth_fano_1</strong> This is a database of regularized quantum periods for one-dimensional Fano manifolds. There is one entry in the database. Each entry in the database is a key-value …
<strong>The database smooth_fano_1</strong> This is a database of regularized quantum periods for one-dimensional Fano manifolds. There is one entry in the database. Each entry in the database is a key-value record with keys and values as described in the paper: <em>Databases of Quantum Periods for Fano Manifolds</em>, Tom Coates and Alexander M. Kasprzyk, 2021. If you make use of this data, please cite the above paper and the DOI for this data: doi:10.5281/zenodo.5708188
<strong>The database smooth_fano_3</strong> This is a database of regularized quantum periods for three-dimensional Fano manifolds. There are 105 entries in the database. Each entry in the database is a key-value …
<strong>The database smooth_fano_3</strong> This is a database of regularized quantum periods for three-dimensional Fano manifolds. There are 105 entries in the database. Each entry in the database is a key-value record with keys and values as described in the paper: <em>Databases of Quantum Periods for Fano Manifolds</em>, Tom Coates and Alexander M. Kasprzyk, 2021. If you make use of this data, please cite the above paper and the DOI for this data: doi:10.5281/zenodo.5708272
<strong>The database smooth_fano_2</strong> This is a database of regularized quantum periods for two-dimensional Fano manifolds. There are ten entries in the database. Each entry in the database is a key-value …
<strong>The database smooth_fano_2</strong> This is a database of regularized quantum periods for two-dimensional Fano manifolds. There are ten entries in the database. Each entry in the database is a key-value record with keys and values as described in the paper: <em>Databases of Quantum Periods for Fano Manifolds</em>, Tom Coates and Alexander M. Kasprzyk, 2021. If you make use of this data, please cite the above paper and the DOI for this data: doi:10.5281/zenodo.5708232
<strong>The database smooth_fano_4</strong> This is a database of regularized quantum periods for four-dimensional Fano manifolds. The database will be updated as new four-dimensional Fano manifolds are discovered and new regularized …
<strong>The database smooth_fano_4</strong> This is a database of regularized quantum periods for four-dimensional Fano manifolds. The database will be updated as new four-dimensional Fano manifolds are discovered and new regularized quantum periods computed. Each entry in the database is a key-value record with keys and values as described in the paper [CK2021]. If you make use of this data, please cite that paper and the DOI for this data: doi:10.5281/zenodo.5708307 <strong>Names</strong> The database describes Fano varieties via names, as follows: Names of Fano manifolds Name Description P1 one-dimensional projective space P2 two-dimensional projective space dP(k) the del Pezzo surface of degree k given by the blow-up of P2 in 9-k points P3 three-dimensional projective space Q3 a quadric hypersurface in four-dimensional projective space B(3,k) the three-dimensional Fano manifold of Picard rank 1, Fano index 2, and degree 8k V(3,k) the three-dimensional Fano manifold of Picard rank 1, Fano index 1, and degree k MM(r,k) the k-th entry in the Mori-Mukai list of three-dimensional Fano manifolds of Picard rank r, ordered as in [CCGK2016]<br> P4 four-dimensional projective space Q4 a quadric hypersurface in five-dimensional projective space FI(4,k) the four-dimensional Fano manifold of Fano index 3 and degree 81k V(4,k) the four-dimensional Fano manifold of Picard rank 1, Fano index 2, and degree 16k MW(4,k) the k-th entry in Table 12.7 of [IP1999] of four-dimensional Fano manifolds of Fano index 2 and Picard rank greater than 1 Obro(4,k) the k-th four-dimensional Fano toric manifold in Obro's classification [O2007] Str(k) the k-th Strangeway manifold in [CGKS2020] CKP(k) the k-th four-dimensional Fano toric complete intersection in [CKP2015] CKK(k) the k-th four-dimensional Fano quiver flag zero locus in Appendix B of [K2019] A name of the form "S1 x S2", where S1 and S2 are names of Fano manifolds X1 and X2, refers to the product manifold X1 x X2. <strong>References</strong> [CCGK2016] <em>Quantum periods for 3-dimensional Fano manifolds</em>; Tom Coates, Alessio Corti, Sergey Galkin, Alexander M. Kasprzyk; Geometry and Topology 20 (2016), no. 1, 103-256. [CGKS2020] <em>Quantum periods for certain four-dimensional Fano manifolds</em>; Tom Coates, Sergey Galkin, Alexander M. Kasprzyk, Andrew Strangeway; Experimental Math. 29 (2020), no. 2, 183-221. [CK2021] <em>Databases of quantum periods for Fano manifolds</em>; Tom Coates, Alexander M. Kasprzyk; 2021. [CKP2015] <em>Four-dimensional Fano toric complete intersections</em>; Tom Coates, Alexander M. Kasprzyk, Thomas Prince; Proc. Royal Society A 471 (2015), no. 2175, 20140704, 14. [IP1999] <em>Fano varieties</em>; V.A. Iskovskikh, Yu. G. Prokhorov; Encyclopaedia Math. Sci. vol. 47, Springer, Berlin, 1999, 1-247. [K2019] <em>Four-dimensional Fano quiver flag zero loci</em>; Elana Kalashnikov; Proc. Royal Society A 275 (2019), no. 2225, 20180791, 23. [O2007] <em>An algorithm for the classification of smooth Fano polytopes</em>; Mikkel Obro; arXiv:0704.0049 [math.CO]; 2007.
We introduce a class of Laurent polynomials, called maximally mutable Laurent polynomials (MMLPs), which we believe correspond under mirror symmetry to Fano varieties. A subclass of these, called rigid, are …
We introduce a class of Laurent polynomials, called maximally mutable Laurent polynomials (MMLPs), which we believe correspond under mirror symmetry to Fano varieties. A subclass of these, called rigid, are expected to correspond to Fano varieties with terminal locally toric singularities. We prove that there are exactly 10 mutation classes of rigid MMLPs in two variables; under mirror symmetry these correspond one-to-one with the 10 deformation classes of smooth del Pezzo surfaces. Furthermore, we give a computer-assisted classification of rigid MMLPs in three variables with reflexive Newton polytope; under mirror symmetry these correspond one-to-one with the 98 deformation classes of three-dimensional Fano manifolds with very ample anti-canonical bundle. We compare our proposal to previous approaches to constructing mirrors to Fano varieties, and explain why mirror symmetry in higher dimensions necessarily involves varieties with terminal singularities. Every known mirror to a Fano manifold, of any dimension, is a rigid MMLP.
We introduce a class of Laurent polynomials, called maximally mutable Laurent polynomials (MMLPs), that we believe correspond under mirror symmetry to Fano varieties. A subclass of these, called rigid, are …
We introduce a class of Laurent polynomials, called maximally mutable Laurent polynomials (MMLPs), that we believe correspond under mirror symmetry to Fano varieties. A subclass of these, called rigid, are expected to correspond to Fano varieties with terminal locally toric singularities. We prove that there are exactly 10 mutation classes of rigid MMLPs in two variables; under mirror symmetry these correspond one-to-one with the 10 deformation classes of smooth del~Pezzo surfaces. Furthermore we give a computer-assisted classification of rigid MMLPs in three variables with reflexive Newton polytope; under mirror symmetry these correspond one-to-one with the 98 deformation classes of three-dimensional Fano manifolds with very ample anticanonical bundle. We compare our proposal to previous approaches to constructing mirrors to Fano varieties, and explain why mirror symmetry in higher dimensions necessarily involves varieties with terminal singularities. Every known mirror to a Fano manifold, of any dimension, is a rigid MMLP.
We give a sharp upper bound on the multiplicity of a fake weighted projective space with at worst canonical singularities. This is equivalent to giving a sharp upper bound on …
We give a sharp upper bound on the multiplicity of a fake weighted projective space with at worst canonical singularities. This is equivalent to giving a sharp upper bound on the index of the sublattice generated by the vertices of a lattice simplex containing only the origin as an interior lattice point. We also completely characterise when equality occurs and discuss related questions and conjectures.
We introduce machine learning methodology to the study of lattice polytopes. With supervised learning techniques, we predict standard properties such as volume, dual volume, reflexivity, etc, with accuracies up to …
We introduce machine learning methodology to the study of lattice polytopes. With supervised learning techniques, we predict standard properties such as volume, dual volume, reflexivity, etc, with accuracies up to 100%. We focus on 2d polygons and 3d polytopes with Pl\"ucker coordinates as input, which out-perform the usual vertex representation.
Fano manifolds are basic building blocks in geometry - they are, in a precise sense, atomic pieces of shapes. The classification of Fano manifolds is therefore an important problem in …
Fano manifolds are basic building blocks in geometry - they are, in a precise sense, atomic pieces of shapes. The classification of Fano manifolds is therefore an important problem in geometry, which has been open since the 1930s. One can think of this as building a Periodic Table for shapes. A recent breakthrough in Fano classification involves a technique from theoretical physics called Mirror Symmetry. From this perspective, a Fano manifold is encoded by a sequence of integers: the coefficients of a power series called the regularized quantum period. Progress to date has been hindered by the fact that quantum periods require specialist expertise to compute, and descriptions of known Fano manifolds and their regularized quantum periods are incomplete and scattered in the literature. We describe databases of regularized quantum periods for Fano manifolds in dimensions up to four. The databases in dimensions one, two, and three are complete; the database in dimension four will be updated as new four-dimensional Fano manifolds are discovered and new regularized quantum periods computed.
The Fine interior $\Delta^{\text{FI}}$ of a $d$-dimensional lattice polytope $\Delta$ is a rational subpolytope of $\Delta$ which is important for constructing minimal birational models of non-degenerate hypersurfaces defined by Laurent …
The Fine interior $\Delta^{\text{FI}}$ of a $d$-dimensional lattice polytope $\Delta$ is a rational subpolytope of $\Delta$ which is important for constructing minimal birational models of non-degenerate hypersurfaces defined by Laurent polynomials with Newton polytope $\Delta$. This paper presents some computational results on the Fine interior of all $674,\!688$ three-dimensional canonical Fano polytopes.
Gorenstein formats present the equations of regular canonical, Calabi–Yau and Fano varieties embedded by subcanonical divisors. We present a new algorithm for the enumeration of these formats based on orbifold …
Gorenstein formats present the equations of regular canonical, Calabi–Yau and Fano varieties embedded by subcanonical divisors. We present a new algorithm for the enumeration of these formats based on orbifold Riemann–Roch and knapsack packing-type algorithms. We apply this to extend the known lists of threefolds of general type beyond the well-known classes of complete intersections and also to find classes of Calabi–Yau threefolds with canonical singularities.
Recent work has focused on the roots $z\in\mathbb{C}$ of the Ehrhart polynomial of a lattice polytope $P$. The case when $\Re{z}=-1/2$ is of particular interest: these polytopes satisfy Golyshev's "canonical …
Recent work has focused on the roots $z\in\mathbb{C}$ of the Ehrhart polynomial of a lattice polytope $P$. The case when $\Re{z}=-1/2$ is of particular interest: these polytopes satisfy Golyshev's "canonical line hypothesis". We characterise such polytopes when $\mathrm{dim}(P)\leq 7$. We also consider the "half-strip condition", where all roots $z$ satisfy $-\mathrm{dim}(P)/2\leq\Re{z}\leq \mathrm{dim}(P)/2-1$, and show that this holds for any reflexive polytope with $\mathrm{dim}(P)\leq 5$. We give an example of a $10$-dimensional reflexive polytope which violates the half-strip condition, thus improving on an example by Ohsugi–Shibata in dimension $34$.
In the paper "Birational geometry via moduli spaces" by I. Cheltsov, L. Katzarkov, and V. Przyjalkowski a new structure connecting toric degenerations of smooth Fano threefolds by projections was introduced; …
In the paper "Birational geometry via moduli spaces" by I. Cheltsov, L. Katzarkov, and V. Przyjalkowski a new structure connecting toric degenerations of smooth Fano threefolds by projections was introduced; using Mirror Symmetry these connections were transferred to the side of Landau--Ginzburg models. In the paper mentioned above a nice way to connect of Picard rank one Fano threefolds was found. We apply this approach to all smooth Fano threefolds, connecting their degenerations by toric basic links. In particular, we find a lot of Gorenstein toric degenerations of smooth Fano threefolds we need. We implement mutations in the picture as well. It turns out that appropriate chosen toric degenerations of the Fanos are given by toric basic links from a few roots. We interpret the relations we found in terms of Mirror Symmetry.
We describe a practical and effective method for reconstructing the deformation class of a Fano manifold X from a Laurent polynomial f that corresponds to X under Mirror Symmetry.We explore …
We describe a practical and effective method for reconstructing the deformation class of a Fano manifold X from a Laurent polynomial f that corresponds to X under Mirror Symmetry.We explore connections to nef partitions, the smoothing of singular toric varieties, and the construction of embeddings of one (possibly-singular) toric variety in another.In particular, we construct degenerations from Fano manifolds to singular toric varieties; in the toric complete intersection case, these degenerations were constructed previously by Doran-Harder.We use our method to find models of orbifold del Pezzo surfaces as complete intersections and degeneracy loci, and to construct a new four-dimensional Fano manifold.
We collect a list of known four-dimensional Fano manifolds and compute their quantum periods. This list includes all four-dimensional Fano manifolds of index greater than one, all four-dimensional toric Fano …
We collect a list of known four-dimensional Fano manifolds and compute their quantum periods. This list includes all four-dimensional Fano manifolds of index greater than one, all four-dimensional toric Fano manifolds, all four-dimensional products of lower-dimensional Fano manifolds, and certain complete intersections in projective bundles.
The Ehrhart quasi-polynomial of a rational polytope $P$ is a fundamental invariant counting lattice points in integer dilates of $P$. The quasi-period of this quasi-polynomial divides the denominator of $P$ …
The Ehrhart quasi-polynomial of a rational polytope $P$ is a fundamental invariant counting lattice points in integer dilates of $P$. The quasi-period of this quasi-polynomial divides the denominator of $P$ but is not always equal to it: this situation is called quasi-period collapse. Polytopes experiencing quasi-period collapse appear widely across algebra and geometry, and yet the phenomenon remains largely mysterious. Using techniques from algebraic geometry - specifically the $\mathbb{Q}$-Gorenstein deformation theory of orbifold del Pezzo surfaces - we explain quasi-period collapse for rational polygons dual to Fano polygons and describe explicitly the discrepancy between the quasi-period and the denominator.
We study $${\mathbb {Q}}$$ -factorial terminal Fano 3-folds whose equations are modelled on those of the Segre embedding of . These lie in codimension 4 in their total anticanonical embedding …
We study $${\mathbb {Q}}$$ -factorial terminal Fano 3-folds whose equations are modelled on those of the Segre embedding of . These lie in codimension 4 in their total anticanonical embedding and have Picard rank 2. They fit into the current state of classification in three different ways. Some families arise as unprojections of degenerations of complete intersections, where the generic unprojection is a known prime Fano 3-fold in codimension 3; these are new, and an analysis of their Gorenstein projections reveals yet other new families. Others represent the "second Tom" unprojection families already known in codimension 4, and we show that every such family contains one of our models. Yet others have no easy Gorenstein projection analysis at all, so prove the existence of Fano components on their Hilbert scheme.
We consider families ${\cal F}(Δ)$ consisting of complex $(n-1)$-dimensional projective algebraic compactifications of $Δ$-regular affine hypersurfaces $Z_f$ defined by Laurent polynomials $f$ with a fixed $n$-dimensional Newton polyhedron $Δ$ in …
We consider families ${\cal F}(Δ)$ consisting of complex $(n-1)$-dimensional projective algebraic compactifications of $Δ$-regular affine hypersurfaces $Z_f$ defined by Laurent polynomials $f$ with a fixed $n$-dimensional Newton polyhedron $Δ$ in $n$-dimensional algebraic torus ${\bf T} =({\bf C}^*)^n$. If the family ${\cal F}(Δ)$ defined by a Newton polyhedron $Δ$ consists of $(n-1)$-dimensional Calabi-Yau varieties, then the dual, or polar, polyhedron $Δ^*$ in the dual space defines another family ${\cal F}(Δ^*)$ of Calabi-Yau varieties, so that we obtain the remarkable duality between two {\em different families} of Calabi-Yau varieties. It is shown that the properties of this duality coincide with the properties of {\em Mirror Symmetry} discovered by physicists for Calabi-Yau $3$-folds. Our method allows to construct many new examples of Calabi-Yau $3$-folds and new candidats for their mirrors which were previously unknown for physicists. We conjecture that there exists an isomorphism between two conformal field theories corresponding to Calabi-Yau varieties from two families ${\cal F}(Δ)$ and ${\cal F}(Δ^*)$.
The quantum period of a variety X is a generating function for certain Gromov-Witten invariants of X which plays an important role in mirror symmetry.We compute the quantum periods of …
The quantum period of a variety X is a generating function for certain Gromov-Witten invariants of X which plays an important role in mirror symmetry.We compute the quantum periods of all 3-dimensional Fano manifolds.In particular we show that 3-dimensional Fano manifolds with very ample anticanonical bundle have mirrors given by a collection of Laurent polynomials called Minkowski polynomials.This was conjectured in joint work with Golyshev.It suggests a new approach to the classification of Fano manifolds: by proving an appropriate mirror theorem and then classifying Fano mirrors.Our methods are likely to be of independent interest.We rework the Mori-Mukai classification of 3-dimensional Fano manifolds, showing that each of them can be expressed as the zero locus of a section of a homogeneous vector bundle over a GIT quotient V = =G , where G is a product of groups of the form GL n .C/ and V is a representation of G .When G D GL 1 .C/ r , this expresses the Fano 3fold as a toric complete intersection; in the remaining cases, it expresses the Fano 3-fold as a tautological subvariety of a Grassmannian, partial flag manifold, or projective bundle thereon.We then compute the quantum periods using the quantum Lefschetz hyperplane theorem of Coates and Givental and the abelian/non-abelian correspondence of Bertram, Ciocan-Fontanine, Kim and Sabbah.
Four dimensional reflexive polyhedra encode the data for smooth Calabi-Yau threefolds that are hypersurfaces in toric varieties, and have important applications both in perturbative and in non-perturbative string theory.We describe …
Four dimensional reflexive polyhedra encode the data for smooth Calabi-Yau threefolds that are hypersurfaces in toric varieties, and have important applications both in perturbative and in non-perturbative string theory.We describe how we obtained all 473,800,776 reflexive polyhedra that exist in four dimensions and the 30,108 distinct pairs of Hodge numbers of the resulting Calabi-Yau manifolds.As a byproduct we show that all these spaces (and hence the corresponding string vacua) are connected via a chain of singular transitions.
An inductive approach to classifying toric Fano varieties is given. As an application of this technique, we present a classification of the toric Fano threefolds with at worst canonical singularities. …
An inductive approach to classifying toric Fano varieties is given. As an application of this technique, we present a classification of the toric Fano threefolds with at worst canonical singularities. Up to isomorphism, there are 674,688 such varieties.
Given a Laurent polynomial f , one can form the period of f : this is a function of one complex variable that plays an important role in mirror symmetry …
Given a Laurent polynomial f , one can form the period of f : this is a function of one complex variable that plays an important role in mirror symmetry for Fano manifolds.Mutations are a particular class of birational transformations acting on Laurent polynomials in two variables; they preserve the period and are closely connected with cluster algebras.We propose a higher-dimensional analog of mutation acting on Laurent polynomials f in n variables.In particular we give a combinatorial description of mutation acting on the Newton polytope P of f , and use this to establish many basic facts about mutations.Mutations can be understood combinatorially in terms of Minkowski rearrangements of slices of P , or in terms of piecewise-linear transformations acting on the dual polytope P * (much like cluster transformations).Mutations map Fano polytopes to Fano polytopes, preserve the Ehrhart series of the dual polytope, and preserve the period of f .Finally we use our results to show that Minkowski polynomials, which are a family of Laurent polynomials that give mirror partners to many three-dimensional Fano manifolds, are connected by a sequence of mutations if and only if they have the same period.
Abstract We classify del Pezzo surfaces with quotient singularities and Picard rank one which admit a ℚ-Gorenstein smoothing. These surfaces arise as singular fibres of del Pezzo fibrations in the …
Abstract We classify del Pezzo surfaces with quotient singularities and Picard rank one which admit a ℚ-Gorenstein smoothing. These surfaces arise as singular fibres of del Pezzo fibrations in the 3-fold minimal model program and also in moduli problems.
A fake weighted projective space X is a Q-factorial toric variety with Picard number one. As with weighted projective space, X comes equipped with a set of weights (λ0, ..., …
A fake weighted projective space X is a Q-factorial toric variety with Picard number one. As with weighted projective space, X comes equipped with a set of weights (λ0, ..., λn). We see how the singularities of P (λ0, ..., λn) influence the singularities of X, and how the weights bound the number of possible fake weighted projective spaces for a fixed dimension. Finally, we present an upper bound on the ratios λj/Σλi if we wish X to have only terminal (or canonical) singularities.
We present an algorithm that produces the classification list of smooth Fano d-polytopes for any given d. The input of the algorithm is a single number, namely the positive integer …
We present an algorithm that produces the classification list of smooth Fano d-polytopes for any given d. The input of the algorithm is a single number, namely the positive integer d. The algorithm has been used to classify smooth Fano d-polytopes for d<=7. There are 7622 isomorphism classes of smooth Fano 6-polytopes and 72256 isomorphism classes of smooth Fano 7-polytopes.
We consider mirror symmetry for Fano manifolds, and describe how one can recover the classification of 3-dimensional Fano manifolds from the study of their mirrors. We sketch a program to …
We consider mirror symmetry for Fano manifolds, and describe how one can recover the classification of 3-dimensional Fano manifolds from the study of their mirrors. We sketch a program to classify 4-dimensional Fano manifolds using these ideas.
We present the last missing details of our algorithm for the classification of reflexive polyhedra in arbitrary dimensions.We also present the results of an application of this algorithm to the …
We present the last missing details of our algorithm for the classification of reflexive polyhedra in arbitrary dimensions.We also present the results of an application of this algorithm to the case of three dimensional reflexive polyhedra.We get 4319 such polyhedra that give rise to K3 surfaces embedded in toric varieties.16 of these contain all others as subpolyhedra.The 4319 polyhedra form a single connected web if we define two polyhedra to be connected if one of them contains the other.
Contents Introduction Chapter I. Affine toric varieties § 1. Cones, lattices, and semigroups § 2. The definition of an affine toric variety § 3. Properties of toric varieties § 4. …
Contents Introduction Chapter I. Affine toric varieties § 1. Cones, lattices, and semigroups § 2. The definition of an affine toric variety § 3. Properties of toric varieties § 4. Differential forms on toric varieties Chapter II. General toric varieties § 5. Fans and their associated toric varieties § 6. Linear systems § 7. The cohomology of invertible sheaves § 8. Resolution of singularities § 9. The fundamental group Chapter III. Intersection theory § 10. The Chow ring § 11. The Riemann-Roch theorem § 12. Complex cohomology Chapter IV. The analytic theory § 13. Toroidal varieties § 14. Quasi-smooth varieties § 15. Differential forms with logarithmic poles Appendix 1. Depth and local cohomology Appendix 2. The exterior algebra Appendix 3. Differentials References
We prove a generalized mirror conjecture for non-negative complete intersections in symplectic toric manifolds. Namely, we express solutions of the PDE system describing quantum cohomology of such a manifold in …
We prove a generalized mirror conjecture for non-negative complete intersections in symplectic toric manifolds. Namely, we express solutions of the PDE system describing quantum cohomology of such a manifold in terms of suitable hypergeometric functions.
We define fake weighted projective spaces as a generalisation of weighted projective spaces. We introduce the notions of fundamental group in codimension 1 and of universal covering in codimension 1. …
We define fake weighted projective spaces as a generalisation of weighted projective spaces. We introduce the notions of fundamental group in codimension 1 and of universal covering in codimension 1. We prove that for every fake weighted projective space its universal cover in codimension 1 is a weighted projective space.
We give a general criterion for two toric varieties to appear as fibers in a flat family over P 1 .We apply this to show that certain birational transformations mapping …
We give a general criterion for two toric varieties to appear as fibers in a flat family over P 1 .We apply this to show that certain birational transformations mapping a Laurent polynomial to another Laurent polynomial correspond to deformations between the associated toric varieties.
<!-- *** Custom HTML *** --> This paper introduces a temporary definition of <i>minimal models</i> of 3-folds (0.7), and studies these under extra hypotheses. The main result is Theorem (0.6), …
<!-- *** Custom HTML *** --> This paper introduces a temporary definition of <i>minimal models</i> of 3-folds (0.7), and studies these under extra hypotheses. The main result is Theorem (0.6), in which I characterise the singularities which necessarily appear on a minimal model, and prove the existence of a minimal model $S$ of a 3-fold of f.g. general type, by blowing up the canonical model $X$ studied in [C3-f], imitating closely the minimal resolution of Du Val surface singularities. Apart from techniques familiar from [C3-f] (computations of the valuations of differentials; cyclic covers; crepant blow-ups of index 1 points which are not cDV), the main new element (Theorem (2.6)) is a method of blowing up the 1-dimensional singular locus, based on the Brieskorn–Tyurina result on the existence of simultaneous resolutions of a family of Du Val surface singularities, together with the elementary transformations in $(-2)$-curves of Burns and Rapoport. Part II is devoted to an exposition of these elementary transformations; much of this is folklore material, but it seems worthwhile to give a detailed account of what seems to be a key phenomenon of higher-dimensional birational geometry. The canonical and terminal singularities introduced in [C3-f] and here have strong inductive properties, and there is some reason for believing that terminal singularities will provide the natural category for an inductive extension of Mori's results: elementary contractions (when these exist) specified by extremal faces of the $K<0$ part of the Mori cone are always discrepant. I have included in §4 conjectures as to what the theory of minimal models and classification of 3-folds will look like in 3 or 4 years' time, and a section of conjectures in §8 attempting to pin down the non-uniqueness of minimal models in the non-ruled case.
We study the geometry of complexified moduli spaces of special Lagrangian submanifolds in the complement of an anticanonical divisor in a compact Kahler manifold. In particular, we explore the connections …
We study the geometry of complexified moduli spaces of special Lagrangian submanifolds in the complement of an anticanonical divisor in a compact Kahler manifold. In particular, we explore the connections between T-duality and mirror symmetry in concrete examples, and show how quantum corrections arise in this context.
For an arbitrary smooth n-dimensional Fano variety $X$ we introduce the notion of a small toric degeneration. Using small toric degenerations of Fano n-folds $X$, we propose a general method …
For an arbitrary smooth n-dimensional Fano variety $X$ we introduce the notion of a small toric degeneration. Using small toric degenerations of Fano n-folds $X$, we propose a general method for constructing mirrors of Calabi-Yau complete intersections in $X$. Our mirror construction is based on a generalized monomial-divisor mirror correspondence which can be used for computing Gromov-Witten invariants of rational curves via specializations of GKZ-hypergeometric series.
A period of a rational integral is the result of integrating, with respect to one or several variables, a rational function over a closed path. This work focuses particularly on …
A period of a rational integral is the result of integrating, with respect to one or several variables, a rational function over a closed path. This work focuses particularly on periods depending on a parameter: in this case the period under consideration satisfies a linear differential equation, the Picard-Fuchs equation. I give a reduction algorithm that extends the Griffiths-Dwork reduction and apply it to the computation of Picard-Fuchs equations. The resulting algorithm is elementary and has been successfully applied to problems that were previously out of reach.
We consider Fano threefolds V with canonical Gorenstein singularities.A sharp bound -K 3V ≤ 72 of the degree is proved.The work was partially supported by grants RFFI 02-01-00441, NS-489.2003.1,NS-1910.2003.1, and …
We consider Fano threefolds V with canonical Gorenstein singularities.A sharp bound -K 3V ≤ 72 of the degree is proved.The work was partially supported by grants RFFI 02-01-00441, NS-489.2003.1,NS-1910.2003.1, and OM RAN.[12, Ch. 4, Remark 4.2]).Weighted projective spaces P(3, 1, 1, 1) and P(6, 4, 1, 1) are Fano threefolds with canonical Gorenstein singularities.Here -K 3 = 72, so they cannot be deformed to smooth ones.The main result of this paper is the following theorem.Theorem 1.5.Let V be a Fano threefold with canonical Gorenstein singularities.Then -K 3 V ≤ 72.Moreover, if the equality -K 3 V = 72 holds, then V is isomorphic to a weighted projective space in Example 1.4.Thus the present paper completely proves Fano-Iskovskikh Conjecture.The following is an immediate consequence of our results.Corollary 1.6.Let V ⊂ P n be a normal projective threefold such that the following condition holds:( †) a general hyperplane sectionIndeed, according to [3] (see also [36]) the variety V is Gorenstein and the anti-canonical class -K V is the class of hyperplane section.If the singularities of V are rational, then they are canonical and by Theorem 1.5 deg V = -K 3 V ≤ 72.If the locus of non-rational singularities is non-empty, then it is zero-dimensional and by the main result of [11] we obtain that V is a cone. Similarly we haveCorollary 1.7.Let U ⊂ P n be a normal projective threefold such that the following condition holds: ( † †) a general hyperplane section U ∩ P n-1 is an Enriques surface with at worst Du Val singularities.If deg U > 36, then U is a cone.Indeed, assume that U is not a cone.Then according to [36] (cf.[4]), the variety U is Q-Gorenstein, has only canonical singularities, and its anticanonical Weil divisor -K U is Q-linearly equivalent to the class of hyperplane section H.This means that n(K U + H) ∼ 0 for some n ∈ N. The divisor K U + H defines an n-sheeted covering π : V → U which is étale over the smooth locus of U. Then the variety V satisfies the conditions of Theorem 1.5 and therefore, -K 3 V ′ ≤ 72.Hence, deg U = -K 3 U = -K 3 V /n ≤ 36.Note that in contrast with Corollary 1.6, the bound in Corollary 1.7 is not sharp.For example it is easy to show that an involution can not act on varieties from Example 1.4 so that the quotient has only canonical singularities and the action is free in codimension two.Therefore, deg U = 36.We give one more consequence of our theorem.Corollary 1.8.Let X be a Fano threefold with canonical (not necessarily Gorenstein) singularities.Assume that
Abstract A lattice polytope is a polytope in whose vertices are all in . The volume of a lattice polytope P containing exactly k ≥ 1 points in d in …
Abstract A lattice polytope is a polytope in whose vertices are all in . The volume of a lattice polytope P containing exactly k ≥ 1 points in d in its interior is bounded above by . Any lattice polytope in of volume V can after an integral unimodular transformation be contained in a lattice cube having side length at most n ˙ n ! V. Thus the number of equivalence classes under integer unimodular transformations of lattice poly topes of bounded volume is finite. If S is any simplex of maximum volume inside a closed bounded convex body K in having nonempty interior, then K ⊆ ( n + 2) S — ( n + l)s where m S denotes a nomothetic copy of S with scale factor m, and s is the centroid of S .
Abstract In previous work by Coates, Galkin and the authors, the notion of mutation between lattice polytopes was introduced. Such mutations give rise to a deformation between the corresponding toric …
Abstract In previous work by Coates, Galkin and the authors, the notion of mutation between lattice polytopes was introduced. Such mutations give rise to a deformation between the corresponding toric varieties. In this paper we study one-step mutations that correspond to deformations between weighted projective planes, giving a complete characterization of such mutations in terms of T -singularities. We also show that the weights involved satisfy Diophantine equations, generalizing results of Hacking and Prokhorov.
We prove mirror symmetry for supersymmetric sigma models on Kahler manifolds in 1+1 dimensions. The proof involves establishing the equivalence of the gauged linear sigma model, embedded in a theory …
We prove mirror symmetry for supersymmetric sigma models on Kahler manifolds in 1+1 dimensions. The proof involves establishing the equivalence of the gauged linear sigma model, embedded in a theory with an enlarged gauge symmetry, with a Landau-Ginzburg theory of Toda type. Standard R -> 1/R duality and dynamical generation of superpotential by vortices are crucial in the derivation. This provides not only a proof of mirror symmetry in the case of (local and global) Calabi-Yau manifolds, but also for sigma models on manifolds with positive first Chern class, including deformations of the action by holomorphic isometries.
We state a number of conjectures that together allow one to classify a broad class of del Pezzo surfaces with cyclic quotient singularities using mirror symmetry. We prove our conjectures …
We state a number of conjectures that together allow one to classify a broad class of del Pezzo surfaces with cyclic quotient singularities using mirror symmetry. We prove our conjectures in the simplest cases. The conjectures relate mutation-equivalence classes of Fano polygons with $\mathbb {Q}$-Gorenstein deformation classes of del Pezzo surfaces.
We present a classification of all weighted projective spaces with at worst terminal or canonical singularities in dimension four. As a corollary we also classify all four-dimensional one-point lattice simplices …
We present a classification of all weighted projective spaces with at worst terminal or canonical singularities in dimension four. As a corollary we also classify all four-dimensional one-point lattice simplices up to equivalence. Finally, we classify the terminal Gorenstein weighted projective spaces up to dimension ten.
It is shown that, for any lattice polytope P⊂ℝd the set int (P)∩lℤd (provided that it is non-empty) contains a point whose coefficient of asymmetry with respect to P is …
It is shown that, for any lattice polytope P⊂ℝd the set int (P)∩lℤd (provided that it is non-empty) contains a point whose coefficient of asymmetry with respect to P is at most . If, moreover, P is a simplex, then this bound can be improved to . As an application, new upper bounds on the volume of a lattice polytope are deduced, given its dimension and the number of sublattice points in its interior.
We find at least 527 new four-dimensional Fano manifolds, each of which is a complete intersection in a smooth toric Fano manifold.
We find at least 527 new four-dimensional Fano manifolds, each of which is a complete intersection in a smooth toric Fano manifold.
We consider mirror symmetry for Fano manifolds, and describe how one can recover the classification of 3-dimensional Fano manifolds from the study of their mirrors. We sketch a program to …
We consider mirror symmetry for Fano manifolds, and describe how one can recover the classification of 3-dimensional Fano manifolds from the study of their mirrors. We sketch a program to classify 4-dimensional Fano manifolds using these ideas.
The purpose of this paper is to give basic tools for the classification of nonsingular toric Fano verieties by means of the notions of primitive collections and primitive relations due …
The purpose of this paper is to give basic tools for the classification of nonsingular toric Fano verieties by means of the notions of primitive collections and primitive relations due to Batyrev. By using them we can easily deal with equivariant blow-ups and blow-downs, and get an easy criterion to determine whether a given nonsingular toric variety is a Fano variety or not. As applications of these results, we get a toric version of a theorem of Mori, and can classify, in principle, all nonsingular toric Fano varieties obtained from a given nonsingular toric Fano variety by finite successions of equivariant blow-ups and blow-downs through nonsingular toric Fano varieties. Especially, we get a new method for the classification of nonsingular toric Fano varieties of dimension at most four. These methods are extended to the case of Gorenstein toric Fano varieties endowed with natural resolutions of singularities. Especially, we easily get a new method for the classification of Gorenstein toric Fano surfaces.
We give an orbifold Riemann-Roch formula in closed form for the Hilbert series of a quasismooth polarized n-fold (X, D), under the assumption that X is projectively Gorenstein with only …
We give an orbifold Riemann-Roch formula in closed form for the Hilbert series of a quasismooth polarized n-fold (X, D), under the assumption that X is projectively Gorenstein with only isolated orbifold points.Our formula is a sum of parts each of which is integral and Gorenstein symmetric of the same canonical weight; the orbifold parts are called ice cream functions.This form of the Hilbert series is particularly useful for computer algebra, and we illustrate it on examples of K3 surfaces and Calabi-Yau 3-folds.These results apply also with higher dimensional orbifold strata (see [8] and [21]), although the correct statements are considerably trickier.We expect to return to this in future publications.
We study Fano threefolds with terminal singularities admitting a "minimal" action of a finite group.We prove that under certain additional assumptions such a variety does not contain planes.We also obtain …
We study Fano threefolds with terminal singularities admitting a "minimal" action of a finite group.We prove that under certain additional assumptions such a variety does not contain planes.We also obtain an upper bounds of the number of singular points of certain Fano threefolds with terminal factorial singularities.1 2 (-K X ) 3 + 1 (see Definition 2.1).We prove the following theorem.Theorem 1.1.Let X be a G-Fano threefold of the main series with g(X) ≥ 6.Then X does not contain any planes.
This paper classifies all toric Fano 3-folds with terminal singularities. This is achieved by solving the equivalent combinatoric problem; that of finding, up to the action of GL(3,Z), all convex …
This paper classifies all toric Fano 3-folds with terminal singularities. This is achieved by solving the equivalent combinatoric problem; that of finding, up to the action of GL(3,Z), all convex polytopes in Z^3 which contain the origin as the only non-vertex lattice point.
We investigate Hodge-theoretic properties of Calabi-Yau complete intersections V of r semi-ample divisors in d-dimensional toric Fano varieties having at most Gorenstein singularities.Our main purpose is to show that the …
We investigate Hodge-theoretic properties of Calabi-Yau complete intersections V of r semi-ample divisors in d-dimensional toric Fano varieties having at most Gorenstein singularities.Our main purpose is to show that the combinatorial duality proposed by second author agrees with the duality for Hodge numbers predicted by mirror symmetry.It is expected that the complete verification of mirror symmetry predictions for singular Calabi-Yau varieties V in arbitrary dimension demands considerations of so called stringtheoretic Hodge numbers h p,q st (V ).We restrict ourselves to the string-theoretic Hodge numbers h 0,q st (V ) and h 1,q st (V ) (0 ≤ q ≤ dr) which coincide with the usual Hodge numbers h 0,q ( V ) and h 1,q ( V ) of a M P CP -desingularization V of V .
In this paper we propose and discuss a mirror construction for complete intersections in partial flag manifolds F (n1, . . . , nl, n). This construction includes our previous …
In this paper we propose and discuss a mirror construction for complete intersections in partial flag manifolds F (n1, . . . , nl, n). This construction includes our previous mirror construction for complete intersection in Grassmannians and the mirror construction of Givental for complete flag manifolds. The key idea of our construction is a degeneration of F (n1, . . . , nl, n) to a certain Gorenstein toric Fano variety P (n1, . . . , nl, n) which has been investigated by Gonciulea and Lakshmibai. We describe a natural small crepant desingularization of P (n1, . . . , nl, n) and prove a generalized version of a conjecture of Gonciulea and Lakshmibai on the singular locus of P (n1, . . . , nl, n). Mathematisches Institut, Eberhard-Karls-Universitat Tubingen, D-72076 Tubingen, Germany, email address: [email protected] Department of Mathematics, Oklahoma State University, Stillwater, OK 74078, USA email address: [email protected] Department of Mathematics, University of California Davis, Davis, CA 95616, USA email address: [email protected] FB 17, Mathematik, Johannes Gutenberg-Universitat Mainz, D-55099 Mainz, Germany, email address: [email protected]
It is pretty well-known that toric Fano varieties of dimension k with terminal singularities correspond to convex lattice polytopes P in R^k of positive finite volume, such that intersection of …
It is pretty well-known that toric Fano varieties of dimension k with terminal singularities correspond to convex lattice polytopes P in R^k of positive finite volume, such that intersection of P and Z^k consists of the point 0 and vertices of P. Likewise, Q-factorial terminal toric singularities essentially correspond to lattice simplexes with no lattice points inside or on the boundary (except the vertices). There have been a lot work, especially in the last 20 years or so on classification of these objects. The main goal of this paper is to bring together these and related results, that are currently scattered in the literature. We also want to emphasize the deep similarity between the problems of classification of toric Fano varieties and classification of Q-factorial toric singularities.