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Mirror symmetry and the classification of orbifold del Pezzo surfaces

2015-09-24
Mohammad Akhtar , Tom Coates , Alessio Corti +6 more
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.
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Homogeneous deformations of toric pairs

2020-06-27
Andrea Petracci
Abstract We extend the Altmann–Mavlyutov construction of homogeneous deformations of affine toric varieties to the case of toric pairs $$(X, \partial X)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>X</mml:mi> <mml:mo>,</mml:mo> <mml:mi>∂</mml:mi> <mml:mi>X</mml:mi> … Abstract We extend the Altmann–Mavlyutov construction of homogeneous deformations of affine toric varieties to the case of toric pairs $$(X, \partial X)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>X</mml:mi> <mml:mo>,</mml:mo> <mml:mi>∂</mml:mi> <mml:mi>X</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> , where X is an affine or projective toric variety and $$\partial X$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>∂</mml:mi> <mml:mi>X</mml:mi> </mml:mrow> </mml:math> is its toric boundary. As an application, we generalise a result due to Ilten to the case of Fano toric pairs.
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On Deformations of Toric Fano Varieties

2022-01-01
Andrea Petracci
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On the quantum periods of del Pezzo surfaces with ⅓ (1, 1) singularities

2018-04-06
Alessandro Oneto , Andrea Petracci
Abstract In earlier joint work with collaborators we gave a conjectural classification of a broad class of orbifold del Pezzo surfaces, using Mirror Symmetry. We proposed that del Pezzo surfaces … Abstract In earlier joint work with collaborators we gave a conjectural classification of a broad class of orbifold del Pezzo surfaces, using Mirror Symmetry. We proposed that del Pezzo surfaces X with isolated cyclic quotient singularities such that X admits a ℚ-Gorenstein toric degeneration correspond via Mirror Symmetry to maximally mutable Laurent polynomials f in two variables, and that the quantum period of such a surface X , which is a generating function for Gromov–Witten invariants of X , coincides with the classical period of its mirror partner f . In this paper we give strong evidence for this conjecture. Contingent on conjectural generalisations of the Quantum Lefschetz theorem and the Abelian/non-Abelian correspondence, we compute many quantum periods for del Pezzo surfaces with <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"><m:mtable><m:mtr><m:mtd><m:mfrac><m:mn>1</m:mn><m:mn>3</m:mn></m:mfrac></m:mtd></m:mtr></m:mtable></m:math> $\begin{array}{} \frac{1}{3} \end{array} $ (1, 1) singularities. Our computations also give strong evidence for the extension of these two principles to the orbifold setting.
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On toric geometry and K-stability of Fano varieties

2021-07-16
Anne-Sophie Kaloghiros , Andrea Petracci
We present some applications of the deformation theory of toric Fano varieties to K-(semi/poly)stability of Fano varieties. First, we present two examples of K-polystable toric Fano 3-fold with obstructed deformations. … We present some applications of the deformation theory of toric Fano varieties to K-(semi/poly)stability of Fano varieties. First, we present two examples of K-polystable toric Fano 3-fold with obstructed deformations. In one case, the K-moduli spaces and stacks are reducible near the closed point associated to the toric Fano 3-fold, while in the other they are non-reduced near the closed point associated to the toric Fano 3-fold. Second, we study K-stability of the general members of two deformation families of smooth Fano 3-folds by building degenerations to K-polystable toric Fano 3-folds.
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Some examples of non-smoothable Gorenstein Fano toric threefolds

2019-08-03
Andrea Petracci
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Mirror Symmetry and Smoothing Gorenstein Toric Affine 3-folds

2022-03-14
Alessio Corti , Matej Filip , Andrea Petracci
A summary is not available for this content so a preview has been provided. Please use the Get access link above for information on how to access this content. A summary is not available for this content so a preview has been provided. Please use the Get access link above for information on how to access this content.
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An Example of Mirror Symmetry for Fano Threefolds

2020-01-01
Andrea Petracci
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On K‐stability of some del Pezzo surfaces of Fano index 2

2022-03-22
Yuchen Liu , Andrea Petracci
For every integer $a \geq 2$, we relate the K-stability of hypersurfaces in the weighted projective space $\mathbb{P}(1,1,a,a)$ of degree $2a$ with the GIT stability of binary forms of degree … For every integer $a \geq 2$, we relate the K-stability of hypersurfaces in the weighted projective space $\mathbb{P}(1,1,a,a)$ of degree $2a$ with the GIT stability of binary forms of degree $2a$. Moreover, we prove that such a hypersurface is K-polystable and not K-stable if it is quasi-smooth.
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The Logarithmic Bogomolov-Tian-Todorov Theorem

2020-10-26
Simon Felten , Andrea Petracci
We prove that the log smooth deformations of a proper log smooth saturated log Calabi-Yau space are unobstructed. We prove that the log smooth deformations of a proper log smooth saturated log Calabi-Yau space are unobstructed.

On deformation spaces of toric singularities and on singularities of K-moduli of Fano varieties

2022-01-26
Andrea Petracci
Firstly, we see that the bases of the miniversal deformations of isolated $\mathbb{Q}$-Gorenstein toric singularities are quite restricted. In particular, we classify the analytic germs of embedding dimension $\leq 2$ … Firstly, we see that the bases of the miniversal deformations of isolated $\mathbb{Q}$-Gorenstein toric singularities are quite restricted. In particular, we classify the analytic germs of embedding dimension $\leq 2$ which are the bases of the miniversal deformations of isolated $\mathbb{Q}$-Gorenstein toric singularities. Secondly, we show that the deformation spaces of isolated Gorenstein toric $3$-fold singularities appear, in a weak sense, as singularities of the K-moduli stack of K-semistable Fano varieties of every dimension $\geq 3$. As a consequence, we prove that the number of local branches of the K-moduli stack of K-semistable Fano varieties and of the K-moduli space of K-polystable Fano varieties is unbounded in each dimension $\geq 3$.
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Mirror Symmetry and the Classification of Orbifold del Pezzo Surfaces

2015-01-21
Mohammad Akhtar , Tom Coates , Alessio Corti +6 more
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 Q-Gorenstein deformation classes of del Pezzo surfaces.

On deformation spaces of toric varieties

2021-05-03
Andrea Petracci
Firstly, we see that the bases of the miniversal deformations of isolated $\mathbb{Q}$-Gorenstein toric singularities are quite restricted. In particular, we classify the analytic germs of embedding dimension $\leq 2$ … Firstly, we see that the bases of the miniversal deformations of isolated $\mathbb{Q}$-Gorenstein toric singularities are quite restricted. In particular, we classify the analytic germs of embedding dimension $\leq 2$ which are the bases of the miniversal deformations of isolated $\mathbb{Q}$-Gorenstein toric singularities. Secondly, we show that the deformation spaces of isolated Gorenstein toric 3-fold singularities appear, in a weak sense, as singularities of the K-moduli stack of K-semistable Fano varieties of every dimension $\geq 3$. As a consequence, we prove that the number of local branches of the K-moduli stack of K-semistable Fano varieties and of the K-moduli space of K-polystable Fano varieties is unbounded in each dimension $\geq 3$.

K-moduli of Fano 3-folds can have embedded points

2021-01-01
Andrea Petracci
We exhibit an example of obstructed K-polystable Fano 3-fold $X$ such that the K-moduli stack of K-semistable Fano varieties and the K-moduli space of K-polystable Fano varieties have an embedded … We exhibit an example of obstructed K-polystable Fano 3-fold $X$ such that the K-moduli stack of K-semistable Fano varieties and the K-moduli space of K-polystable Fano varieties have an embedded point at $[X]$.

The logarithmic Bogomolov–Tian–Todorov theorem

2022-04-01
Simon Felten , Andrea Petracci
We prove that the log smooth deformations of a proper log smooth saturated log Calabi–Yau space are unobstructed. We prove that the log smooth deformations of a proper log smooth saturated log Calabi–Yau space are unobstructed.
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On deformations of toric Fano varieties

2019-12-03
Andrea Petracci
In this note we collect some results on the deformation theory of toric Fano varieties. In this note we collect some results on the deformation theory of toric Fano varieties.
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Some examples of non-smoothable Gorenstein Fano toric threefolds

2018-04-21
Andrea Petracci
We present a combinatorial criterion on reflexive polytopes of dimension 3 which gives a local-to-global obstruction for the smoothability of the corresponding Fano toric threefolds. As a result, we show … We present a combinatorial criterion on reflexive polytopes of dimension 3 which gives a local-to-global obstruction for the smoothability of the corresponding Fano toric threefolds. As a result, we show examples of singular Gorenstein Fano toric threefolds which have compound Du Val, hence smoothable, singularities but are not smoothable.

Deformations of log Calabi-Yau pairs can be obstructed

2021-01-01
Simon Felten , Andrea Petracci , Sharon Robins
We exhibit examples of pairs $(X,D)$ where $X$ is a smooth projective variety and $D$ is an anticanonical reduced simple normal crossing divisor such that the deformations of $(X,D)$ are … We exhibit examples of pairs $(X,D)$ where $X$ is a smooth projective variety and $D$ is an anticanonical reduced simple normal crossing divisor such that the deformations of $(X,D)$ are obstructed. These examples are constructed via toric geometry.
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Reflexive polygons and rational elliptic surfaces

2023-07-12
Antonella Grassi , Giulia Gugiatti , Wendelin Lutz +1 more
Abstract In this note we study in detail the geometry of eight rational elliptic surfaces naturally associated to the sixteen reflexive polygons. The elliptic fibrations supported by these surfaces correspond … Abstract In this note we study in detail the geometry of eight rational elliptic surfaces naturally associated to the sixteen reflexive polygons. The elliptic fibrations supported by these surfaces correspond under mirror symmetry to the eight families of smooth del Pezzo surfaces with very ample anticanonical bundle.
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A 1-Dimensional Component of K-Moduli of del Pezzo Surfaces

2023-01-01
Andrea Petracci

Deformations of log Calabi–Yau pairs can be obstructed

2023-01-01
Simon Felten , Andrea Petracci , Sharon Robins
We exhibit examples of pairs (X, D) where X is a smooth projective variety and D is an anticanonical reduced simple normal crossing divisor such that the deformations of (X, … We exhibit examples of pairs (X, D) where X is a smooth projective variety and D is an anticanonical reduced simple normal crossing divisor such that the deformations of (X, D) are obstructed.These examples are constructed via toric geometry.
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On deformation spaces of toric singularities and on singularities of K-moduli of Fano varieties

2021-05-03
Andrea Petracci
Firstly, we see that the bases of the miniversal deformations of isolated $\mathbb{Q}$-Gorenstein toric singularities are quite restricted. In particular, we classify the analytic germs of embedding dimension $\leq 2$ … Firstly, we see that the bases of the miniversal deformations of isolated $\mathbb{Q}$-Gorenstein toric singularities are quite restricted. In particular, we classify the analytic germs of embedding dimension $\leq 2$ which are the bases of the miniversal deformations of isolated $\mathbb{Q}$-Gorenstein toric singularities. Secondly, we show that the deformation spaces of isolated Gorenstein toric $3$-fold singularities appear, in a weak sense, as singularities of the K-moduli stack of K-semistable Fano varieties of every dimension $\geq 3$. As a consequence, we prove that the number of local branches of the K-moduli stack of K-semistable Fano varieties and of the K-moduli space of K-polystable Fano varieties is unbounded in each dimension $\geq 3$.
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A 1-dimensional component of K-moduli of del Pezzo surfaces

2021-01-01
Andrea Petracci
We explicitly construct a component of the K-moduli space of K-polystable del Pezzo surfaces which is a smooth rational curve. We explicitly construct a component of the K-moduli space of K-polystable del Pezzo surfaces which is a smooth rational curve.
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The logarithmic Bogomolov-Tian-Todorov theorem

2020-01-01
Simon Felten , Andrea Petracci
We prove that the log smooth deformations of a proper log smooth saturated log Calabi-Yau space are unobstructed. We prove that the log smooth deformations of a proper log smooth saturated log Calabi-Yau space are unobstructed.
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An example of Mirror Symmetry for Fano threefolds

2019-01-01
Andrea Petracci
In this note we illustrate the Fanosearch programme of Coates, Corti, Galkin, Golyshev, and Kasprzyk in the example of the anticanonical cone over the smooth del Pezzo surface of degree … In this note we illustrate the Fanosearch programme of Coates, Corti, Galkin, Golyshev, and Kasprzyk in the example of the anticanonical cone over the smooth del Pezzo surface of degree 6.
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Mirror Symmetry and the Classification of Orbifold del Pezzo Surfaces

2015-01-01
Mohammad Akhtar , Thomas D. Coates , Alessio Corti +6 more
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 Q-Gorenstein deformation classes of del Pezzo surfaces.
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On K-moduli of quartic threefolds

2022-01-01
Hamid Abban , Ivan Cheltsov , Alexander Kasprzyk +1 more
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 $X_{2,2,4} \subset \mathbb{P}(1^5,2^2)$.
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Reflexive polygons and rational elliptic surfaces

2023-01-01
Antonella Grassi , Giulia Gugiatti , Wendelin Lutz +1 more
In this note we study in detail the geometry of eight rational elliptic surfaces naturally associated to the sixteen reflexive polygons. The elliptic fibrations supported by these surfaces correspond under … In this note we study in detail the geometry of eight rational elliptic surfaces naturally associated to the sixteen reflexive polygons. The elliptic fibrations supported by these surfaces correspond under mirror symmetry to the eight families of smooth del Pezzo surfaces with very ample anticanonical bundle.
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On K-moduli of Fano threefolds with degree 28 and Picard rank 4

2023-01-01
Liana Heuberger , Andrea Petracci
We analyse the local structure of the K-moduli space of Fano varieties at a toric singular K-polystable Fano 3-fold, which deforms to smooth Fano 3-folds with anticanonical volume 28 and … We analyse the local structure of the K-moduli space of Fano varieties at a toric singular K-polystable Fano 3-fold, which deforms to smooth Fano 3-folds with anticanonical volume 28 and Picard rank 4. In particular, by constructing an algebraic deformation of this toric singular Fano, we show that the irreducible component of K-moduli parametrising these smooth Fano 3-folds is a rational surface.

Smoothing Gorenstein toric Fano 3-folds

2024-12-09
Alessio Corti , Paul Hacking , Andrea Petracci
We introduce admissible Minkowski decomposition data (amd) for a 3-dimensional reflexive polytope P. This notion is defined purely in terms of the combinatorics of P. Denoting by X the Gorenstein … We introduce admissible Minkowski decomposition data (amd) for a 3-dimensional reflexive polytope P. This notion is defined purely in terms of the combinatorics of P. Denoting by X the Gorenstein toric Fano 3-fold whose fan is the spanning fan (a.k.a. face fan) of P, our first result states that amd for P determine a smoothing of X. Our second result amounts to an effective recipe for computing the Betti numbers of the smoothing.
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On K-moduli of quartic threefolds

2025-05-01
Hamid Abban , Ivan Cheltsov , Alexander Kasprzyk +2 more
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On K-moduli of quartic threefolds

2025-05-01
Hamid Abban , Ivan Cheltsov , Alexander Kasprzyk +2 more
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Smoothing Gorenstein toric Fano 3-folds

2024-12-09
Alessio Corti , Paul Hacking , Andrea Petracci
We introduce admissible Minkowski decomposition data (amd) for a 3-dimensional reflexive polytope P. This notion is defined purely in terms of the combinatorics of P. Denoting by X the Gorenstein … We introduce admissible Minkowski decomposition data (amd) for a 3-dimensional reflexive polytope P. This notion is defined purely in terms of the combinatorics of P. Denoting by X the Gorenstein toric Fano 3-fold whose fan is the spanning fan (a.k.a. face fan) of P, our first result states that amd for P determine a smoothing of X. Our second result amounts to an effective recipe for computing the Betti numbers of the smoothing.
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Reflexive polygons and rational elliptic surfaces

2023-07-12
Antonella Grassi , Giulia Gugiatti , Wendelin Lutz +1 more
Abstract In this note we study in detail the geometry of eight rational elliptic surfaces naturally associated to the sixteen reflexive polygons. The elliptic fibrations supported by these surfaces correspond … Abstract In this note we study in detail the geometry of eight rational elliptic surfaces naturally associated to the sixteen reflexive polygons. The elliptic fibrations supported by these surfaces correspond under mirror symmetry to the eight families of smooth del Pezzo surfaces with very ample anticanonical bundle.
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A 1-Dimensional Component of K-Moduli of del Pezzo Surfaces

2023-01-01
Andrea Petracci

Deformations of log Calabi–Yau pairs can be obstructed

2023-01-01
Simon Felten , Andrea Petracci , Sharon Robins
We exhibit examples of pairs (X, D) where X is a smooth projective variety and D is an anticanonical reduced simple normal crossing divisor such that the deformations of (X, … We exhibit examples of pairs (X, D) where X is a smooth projective variety and D is an anticanonical reduced simple normal crossing divisor such that the deformations of (X, D) are obstructed.These examples are constructed via toric geometry.
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Reflexive polygons and rational elliptic surfaces

2023-01-01
Antonella Grassi , Giulia Gugiatti , Wendelin Lutz +1 more
In this note we study in detail the geometry of eight rational elliptic surfaces naturally associated to the sixteen reflexive polygons. The elliptic fibrations supported by these surfaces correspond under … In this note we study in detail the geometry of eight rational elliptic surfaces naturally associated to the sixteen reflexive polygons. The elliptic fibrations supported by these surfaces correspond under mirror symmetry to the eight families of smooth del Pezzo surfaces with very ample anticanonical bundle.
View PDF Chat

On K-moduli of Fano threefolds with degree 28 and Picard rank 4

2023-01-01
Liana Heuberger , Andrea Petracci
We analyse the local structure of the K-moduli space of Fano varieties at a toric singular K-polystable Fano 3-fold, which deforms to smooth Fano 3-folds with anticanonical volume 28 and … We analyse the local structure of the K-moduli space of Fano varieties at a toric singular K-polystable Fano 3-fold, which deforms to smooth Fano 3-folds with anticanonical volume 28 and Picard rank 4. In particular, by constructing an algebraic deformation of this toric singular Fano, we show that the irreducible component of K-moduli parametrising these smooth Fano 3-folds is a rational surface.

The logarithmic Bogomolov–Tian–Todorov theorem

2022-04-01
Simon Felten , Andrea Petracci
We prove that the log smooth deformations of a proper log smooth saturated log Calabi–Yau space are unobstructed. We prove that the log smooth deformations of a proper log smooth saturated log Calabi–Yau space are unobstructed.
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On K‐stability of some del Pezzo surfaces of Fano index 2

2022-03-22
Yuchen Liu , Andrea Petracci
For every integer $a \geq 2$, we relate the K-stability of hypersurfaces in the weighted projective space $\mathbb{P}(1,1,a,a)$ of degree $2a$ with the GIT stability of binary forms of degree … For every integer $a \geq 2$, we relate the K-stability of hypersurfaces in the weighted projective space $\mathbb{P}(1,1,a,a)$ of degree $2a$ with the GIT stability of binary forms of degree $2a$. Moreover, we prove that such a hypersurface is K-polystable and not K-stable if it is quasi-smooth.
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Mirror Symmetry and Smoothing Gorenstein Toric Affine 3-folds

2022-03-14
Alessio Corti , Matej Filip , Andrea Petracci
A summary is not available for this content so a preview has been provided. Please use the Get access link above for information on how to access this content. A summary is not available for this content so a preview has been provided. Please use the Get access link above for information on how to access this content.
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On deformation spaces of toric singularities and on singularities of K-moduli of Fano varieties

2022-01-26
Andrea Petracci
Firstly, we see that the bases of the miniversal deformations of isolated $\mathbb{Q}$-Gorenstein toric singularities are quite restricted. In particular, we classify the analytic germs of embedding dimension $\leq 2$ … Firstly, we see that the bases of the miniversal deformations of isolated $\mathbb{Q}$-Gorenstein toric singularities are quite restricted. In particular, we classify the analytic germs of embedding dimension $\leq 2$ which are the bases of the miniversal deformations of isolated $\mathbb{Q}$-Gorenstein toric singularities. Secondly, we show that the deformation spaces of isolated Gorenstein toric $3$-fold singularities appear, in a weak sense, as singularities of the K-moduli stack of K-semistable Fano varieties of every dimension $\geq 3$. As a consequence, we prove that the number of local branches of the K-moduli stack of K-semistable Fano varieties and of the K-moduli space of K-polystable Fano varieties is unbounded in each dimension $\geq 3$.
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On Deformations of Toric Fano Varieties

2022-01-01
Andrea Petracci
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On K-moduli of quartic threefolds

2022-01-01
Hamid Abban , Ivan Cheltsov , Alexander Kasprzyk +1 more
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 $X_{2,2,4} \subset \mathbb{P}(1^5,2^2)$.
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On toric geometry and K-stability of Fano varieties

2021-07-16
Anne-Sophie Kaloghiros , Andrea Petracci
We present some applications of the deformation theory of toric Fano varieties to K-(semi/poly)stability of Fano varieties. First, we present two examples of K-polystable toric Fano 3-fold with obstructed deformations. … We present some applications of the deformation theory of toric Fano varieties to K-(semi/poly)stability of Fano varieties. First, we present two examples of K-polystable toric Fano 3-fold with obstructed deformations. In one case, the K-moduli spaces and stacks are reducible near the closed point associated to the toric Fano 3-fold, while in the other they are non-reduced near the closed point associated to the toric Fano 3-fold. Second, we study K-stability of the general members of two deformation families of smooth Fano 3-folds by building degenerations to K-polystable toric Fano 3-folds.
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On deformation spaces of toric varieties

2021-05-03
Andrea Petracci
Firstly, we see that the bases of the miniversal deformations of isolated $\mathbb{Q}$-Gorenstein toric singularities are quite restricted. In particular, we classify the analytic germs of embedding dimension $\leq 2$ … Firstly, we see that the bases of the miniversal deformations of isolated $\mathbb{Q}$-Gorenstein toric singularities are quite restricted. In particular, we classify the analytic germs of embedding dimension $\leq 2$ which are the bases of the miniversal deformations of isolated $\mathbb{Q}$-Gorenstein toric singularities. Secondly, we show that the deformation spaces of isolated Gorenstein toric 3-fold singularities appear, in a weak sense, as singularities of the K-moduli stack of K-semistable Fano varieties of every dimension $\geq 3$. As a consequence, we prove that the number of local branches of the K-moduli stack of K-semistable Fano varieties and of the K-moduli space of K-polystable Fano varieties is unbounded in each dimension $\geq 3$.

On deformation spaces of toric singularities and on singularities of K-moduli of Fano varieties

2021-05-03
Andrea Petracci
Firstly, we see that the bases of the miniversal deformations of isolated $\mathbb{Q}$-Gorenstein toric singularities are quite restricted. In particular, we classify the analytic germs of embedding dimension $\leq 2$ … Firstly, we see that the bases of the miniversal deformations of isolated $\mathbb{Q}$-Gorenstein toric singularities are quite restricted. In particular, we classify the analytic germs of embedding dimension $\leq 2$ which are the bases of the miniversal deformations of isolated $\mathbb{Q}$-Gorenstein toric singularities. Secondly, we show that the deformation spaces of isolated Gorenstein toric $3$-fold singularities appear, in a weak sense, as singularities of the K-moduli stack of K-semistable Fano varieties of every dimension $\geq 3$. As a consequence, we prove that the number of local branches of the K-moduli stack of K-semistable Fano varieties and of the K-moduli space of K-polystable Fano varieties is unbounded in each dimension $\geq 3$.
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K-moduli of Fano 3-folds can have embedded points

2021-01-01
Andrea Petracci
We exhibit an example of obstructed K-polystable Fano 3-fold $X$ such that the K-moduli stack of K-semistable Fano varieties and the K-moduli space of K-polystable Fano varieties have an embedded … We exhibit an example of obstructed K-polystable Fano 3-fold $X$ such that the K-moduli stack of K-semistable Fano varieties and the K-moduli space of K-polystable Fano varieties have an embedded point at $[X]$.

Deformations of log Calabi-Yau pairs can be obstructed

2021-01-01
Simon Felten , Andrea Petracci , Sharon Robins
We exhibit examples of pairs $(X,D)$ where $X$ is a smooth projective variety and $D$ is an anticanonical reduced simple normal crossing divisor such that the deformations of $(X,D)$ are … We exhibit examples of pairs $(X,D)$ where $X$ is a smooth projective variety and $D$ is an anticanonical reduced simple normal crossing divisor such that the deformations of $(X,D)$ are obstructed. These examples are constructed via toric geometry.
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A 1-dimensional component of K-moduli of del Pezzo surfaces

2021-01-01
Andrea Petracci
We explicitly construct a component of the K-moduli space of K-polystable del Pezzo surfaces which is a smooth rational curve. We explicitly construct a component of the K-moduli space of K-polystable del Pezzo surfaces which is a smooth rational curve.
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The Logarithmic Bogomolov-Tian-Todorov Theorem

2020-10-26
Simon Felten , Andrea Petracci
We prove that the log smooth deformations of a proper log smooth saturated log Calabi-Yau space are unobstructed. We prove that the log smooth deformations of a proper log smooth saturated log Calabi-Yau space are unobstructed.

Homogeneous deformations of toric pairs

2020-06-27
Andrea Petracci
Abstract We extend the Altmann–Mavlyutov construction of homogeneous deformations of affine toric varieties to the case of toric pairs $$(X, \partial X)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>X</mml:mi> <mml:mo>,</mml:mo> <mml:mi>∂</mml:mi> <mml:mi>X</mml:mi> … Abstract We extend the Altmann–Mavlyutov construction of homogeneous deformations of affine toric varieties to the case of toric pairs $$(X, \partial X)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>X</mml:mi> <mml:mo>,</mml:mo> <mml:mi>∂</mml:mi> <mml:mi>X</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> , where X is an affine or projective toric variety and $$\partial X$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>∂</mml:mi> <mml:mi>X</mml:mi> </mml:mrow> </mml:math> is its toric boundary. As an application, we generalise a result due to Ilten to the case of Fano toric pairs.
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An Example of Mirror Symmetry for Fano Threefolds

2020-01-01
Andrea Petracci
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The logarithmic Bogomolov-Tian-Todorov theorem

2020-01-01
Simon Felten , Andrea Petracci
We prove that the log smooth deformations of a proper log smooth saturated log Calabi-Yau space are unobstructed. We prove that the log smooth deformations of a proper log smooth saturated log Calabi-Yau space are unobstructed.
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On deformations of toric Fano varieties

2019-12-03
Andrea Petracci
In this note we collect some results on the deformation theory of toric Fano varieties. In this note we collect some results on the deformation theory of toric Fano varieties.
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Some examples of non-smoothable Gorenstein Fano toric threefolds

2019-08-03
Andrea Petracci
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An example of Mirror Symmetry for Fano threefolds

2019-01-01
Andrea Petracci
In this note we illustrate the Fanosearch programme of Coates, Corti, Galkin, Golyshev, and Kasprzyk in the example of the anticanonical cone over the smooth del Pezzo surface of degree … In this note we illustrate the Fanosearch programme of Coates, Corti, Galkin, Golyshev, and Kasprzyk in the example of the anticanonical cone over the smooth del Pezzo surface of degree 6.
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Some examples of non-smoothable Gorenstein Fano toric threefolds

2018-04-21
Andrea Petracci
We present a combinatorial criterion on reflexive polytopes of dimension 3 which gives a local-to-global obstruction for the smoothability of the corresponding Fano toric threefolds. As a result, we show … We present a combinatorial criterion on reflexive polytopes of dimension 3 which gives a local-to-global obstruction for the smoothability of the corresponding Fano toric threefolds. As a result, we show examples of singular Gorenstein Fano toric threefolds which have compound Du Val, hence smoothable, singularities but are not smoothable.

On the quantum periods of del Pezzo surfaces with ⅓ (1, 1) singularities

2018-04-06
Alessandro Oneto , Andrea Petracci
Abstract In earlier joint work with collaborators we gave a conjectural classification of a broad class of orbifold del Pezzo surfaces, using Mirror Symmetry. We proposed that del Pezzo surfaces … Abstract In earlier joint work with collaborators we gave a conjectural classification of a broad class of orbifold del Pezzo surfaces, using Mirror Symmetry. We proposed that del Pezzo surfaces X with isolated cyclic quotient singularities such that X admits a ℚ-Gorenstein toric degeneration correspond via Mirror Symmetry to maximally mutable Laurent polynomials f in two variables, and that the quantum period of such a surface X , which is a generating function for Gromov–Witten invariants of X , coincides with the classical period of its mirror partner f . In this paper we give strong evidence for this conjecture. Contingent on conjectural generalisations of the Quantum Lefschetz theorem and the Abelian/non-Abelian correspondence, we compute many quantum periods for del Pezzo surfaces with <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"><m:mtable><m:mtr><m:mtd><m:mfrac><m:mn>1</m:mn><m:mn>3</m:mn></m:mfrac></m:mtd></m:mtr></m:mtable></m:math> $\begin{array}{} \frac{1}{3} \end{array} $ (1, 1) singularities. Our computations also give strong evidence for the extension of these two principles to the orbifold setting.
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Mirror symmetry and the classification of orbifold del Pezzo surfaces

2015-09-24
Mohammad Akhtar , Tom Coates , Alessio Corti +6 more
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.
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Mirror Symmetry and the Classification of Orbifold del Pezzo Surfaces

2015-01-21
Mohammad Akhtar , Tom Coates , Alessio Corti +6 more
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 Q-Gorenstein deformation classes of del Pezzo surfaces.

Mirror Symmetry and the Classification of Orbifold del Pezzo Surfaces

2015-01-01
Mohammad Akhtar , Thomas D. Coates , Alessio Corti +6 more
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 Q-Gorenstein deformation classes of del Pezzo surfaces.
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Coauthor Papers Together
Alessio Corti 5
Simon Felten 5
Alexander Kasprzyk 5
Liana Heuberger 4
Alessandro Oneto 4
Thomas Prince 3
Ketil Tveiten 3
Mohammad Akhtar 2
Wendelin Lutz 2
Yuchen Liu 2
Tom Coates 2
Hamid Abban 2
Giulia Gugiatti 2
Antonella Grassi 2
Sharon Robins 2
Ivan Cheltsov 2
Matej Filip 1
Paul Hacking 1
Anne-Sophie Kaloghiros 1
Thomas D. Coates 1
Mohammad Akhtar 1

Mirror symmetry and the classification of orbifold del Pezzo surfaces

2015-09-24
Mohammad Akhtar , Tom Coates , Alessio Corti +6 more
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.
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Minkowski Polynomials and Mutations

2012-12-08
Mohammad Akhtar
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.
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Mirror Symmetry and Fano Manifolds

2014-01-03
Tom Coates , Alessio Corti , Sergey Galkin +2 more
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.

Mirror Symmetry and Fano Manifolds

2013-11-14
Tom Coates , Alessio Corti , Sergey Galkin +2 more
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.
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Quantum periods for 3–dimensional Fano manifolds

2016-02-29
Tom Coates , Alessio Corti , Sergey Galkin +1 more
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.
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Homogeneous deformations of toric pairs

2020-06-27
Andrea Petracci
Abstract We extend the Altmann–Mavlyutov construction of homogeneous deformations of affine toric varieties to the case of toric pairs $$(X, \partial X)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>X</mml:mi> <mml:mo>,</mml:mo> <mml:mi>∂</mml:mi> <mml:mi>X</mml:mi> … Abstract We extend the Altmann–Mavlyutov construction of homogeneous deformations of affine toric varieties to the case of toric pairs $$(X, \partial X)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>X</mml:mi> <mml:mo>,</mml:mo> <mml:mi>∂</mml:mi> <mml:mi>X</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> , where X is an affine or projective toric variety and $$\partial X$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>∂</mml:mi> <mml:mi>X</mml:mi> </mml:mrow> </mml:math> is its toric boundary. As an application, we generalise a result due to Ilten to the case of Fano toric pairs.
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Jumping of the nef cone for Fano varieties

2011-06-29
Burt Totaro
We construct <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="bold upper Q"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mtext mathvariant="bold">Q</mml:mtext> </mml:mrow> <mml:annotation encoding="application/x-tex">\textbf {Q}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-factorial terminal Fano varieties, starting in dimension 4, whose nef … We construct <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="bold upper Q"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mtext mathvariant="bold">Q</mml:mtext> </mml:mrow> <mml:annotation encoding="application/x-tex">\textbf {Q}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-factorial terminal Fano varieties, starting in dimension 4, whose nef cone jumps when the variety is deformed. It follows that de Fernex and Hacon’s results on deformations of 3-dimensional Fanos are optimal. The examples are based on the existence of high-dimensional flips which deform to isomorphisms, generalizing the Mukai flop. We also improve earlier results on deformations of Fano varieties. Toric Fano varieties which are smooth in codimension 2 and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="bold upper Q"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mtext mathvariant="bold">Q</mml:mtext> </mml:mrow> <mml:annotation encoding="application/x-tex">\textbf {Q}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-factorial in codimension 3 are rigid. The divisor class group is deformation-invariant for klt Fanos which are smooth in codimension 2 and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="bold upper Q"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mtext mathvariant="bold">Q</mml:mtext> </mml:mrow> <mml:annotation encoding="application/x-tex">\textbf {Q}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-factorial in codimension 3. The Cox ring deforms in a flat family under deformation of a terminal Fano which is <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="bold upper Q"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mtext mathvariant="bold">Q</mml:mtext> </mml:mrow> <mml:annotation encoding="application/x-tex">\textbf {Q}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-factorial in codimension 3. A side result which seems to be new is that the divisor class group of a klt Fano variety maps isomorphically to ordinary homology.
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The versal deformation of an isolated toric Gorenstein singularity

1997-09-01
Klaus Altmann
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Minkowski sums and homogeneous deformations of toric varieties

1995-01-01
Klaus Altmann
We investigate those deformations of affine toric varieties (toric singularities) that arise from embedding them into higher dimensional toric varieties as a relative complete intersection.On the one hand, many examples … We investigate those deformations of affine toric varieties (toric singularities) that arise from embedding them into higher dimensional toric varieties as a relative complete intersection.On the one hand, many examples promise that these so-called toric deformations cover a great deal of the entire deformation theory.On the other hand, they can be described explicitly.Toric deformations are related to decompositions (into a Minkowski sum) of cross cuts of the polyhedral cone defining the toric singularity.Finally, we consider the special case of toric Gorenstein singularities.Many of them turn out to be rigid; for the remaining examples the description of their toric deformations becomes easier than in the general case.
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Deformations of Toric Varieties via Minkowski Sum Decompositions of Polyhedral Complexes

2009-01-01
Anvar R. Mavlyutov
We generalized the construction of deformations of affine toric varieties of K. Altmann and our previous construction of deformations of weak Fano toric varieties to the case of arbitrary toric … We generalized the construction of deformations of affine toric varieties of K. Altmann and our previous construction of deformations of weak Fano toric varieties to the case of arbitrary toric varieties by introducing the notion of Minkowski sum decompositions of polyhedral complexes. Our construction embeds the original toric variety into a higher dimensional toric variety where the image is given by a prime binomial complete intersection ideal in Cox homogeneous coordinates. The deformations are realized by families of complete intersections. For compact simplicial toric varieties with at worst Gorenstein terminal singularities, we show that our deformations span the infinitesimal space of deformations by Kodaira-Spencer map. For Fano toric varieties, we show that their deformations can be constructed in higher-dimensional Fano toric varieties related to the Batyrev-Borisov mirror symmetry construction.

Threefolds and deformations of surface singularities

1988-06-01
J. Koll�r , N. I. Shepherd‐Barron

K-polystability of Q-Fano varieties admitting Kähler-Einstein metrics

2015-06-22
Robert J. Berman
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Uniqueness of K-polystable degenerations of Fano varieties

2019-08-22
Harold Blum , Chenyang Xu
We prove that K-polystable degenerations of $\mathbb{Q}$-Fano varieties are unique. Furthermore, we show that the moduli stack of K-stable $\mathbb{Q}$-Fano varieties is separated. Together with recently proven boundedness and openness … We prove that K-polystable degenerations of $\mathbb{Q}$-Fano varieties are unique. Furthermore, we show that the moduli stack of K-stable $\mathbb{Q}$-Fano varieties is separated. Together with recently proven boundedness and openness statements, the latter result yields a separated Deligne-Mumford stack parametrizing all uniformly K-stable $\mathbb{Q}$-Fano varieties of fixed dimension and volume. The result also implies that the automorphism group of a K-stable $\mathbb{Q}$-Fano variety is finite.
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Boundedness of Q-Fano varieties with degrees and alpha-invariants bounded from below

2020-01-01
Chen Jiang
We show that $\mathbb{Q}$-Fano varieties of fixed dimension with anti-canonical degrees and alpha-invariants bounded from below form a bounded family. As a corollary, K-semistable $\mathbb{Q}$-Fano varieties of fixed dimension with … We show that $\mathbb{Q}$-Fano varieties of fixed dimension with anti-canonical degrees and alpha-invariants bounded from below form a bounded family. As a corollary, K-semistable $\mathbb{Q}$-Fano varieties of fixed dimension with anti-canonical degrees bounded from below form a bounded family.
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A Luna étale slice theorem for algebraic stacks

2020-04-15
Jarod Alper , Jack Hall , David Rydh
We prove that every algebraic stack, locally of finite type over an algebraically closed field with affine stabilizers, is étale-locally a quotient stack in a neighborhood of a point with … We prove that every algebraic stack, locally of finite type over an algebraically closed field with affine stabilizers, is étale-locally a quotient stack in a neighborhood of a point with a linearly reductive stabilizer group. The proof uses an equivariant version of Artin's algebraization theorem proved in the appendix. We provide numerous applications of the main theorems.
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Mutations of Laurent Polynomials and Flat Families with Toric Fibers

2012-07-28
Nathan Ilten
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.
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Mirror symmetry for log Calabi-Yau surfaces I

2015-03-24
Mark Gross , Paul Hacking , Seán Keel
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On Deformations of Toric Fano Varieties

2022-01-01
Andrea Petracci
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Differential graded Lie algebras and formal deformation theory

2009-01-01
Marco Manetti

Deformations of Algebraic Schemes

2006-01-01
Edoardo Sernesi

Openness of K-semistability for Fano varieties

2022-01-01
Harold Blum , Yuchen Liu , Chenyang Xu
In this paper, we prove the openness of K-semistability in families of log Fano pairs by showing that the stability threshold is a constructible function on the fibers. We also … In this paper, we prove the openness of K-semistability in families of log Fano pairs by showing that the stability threshold is a constructible function on the fibers. We also prove that any special test configuration arises from a log canonical place of a bounded complement and establish properties of any minimizer of the stability threshold.
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On toric geometry and K-stability of Fano varieties

2021-07-16
Anne-Sophie Kaloghiros , Andrea Petracci
We present some applications of the deformation theory of toric Fano varieties to K-(semi/poly)stability of Fano varieties. First, we present two examples of K-polystable toric Fano 3-fold with obstructed deformations. … We present some applications of the deformation theory of toric Fano varieties to K-(semi/poly)stability of Fano varieties. First, we present two examples of K-polystable toric Fano 3-fold with obstructed deformations. In one case, the K-moduli spaces and stacks are reducible near the closed point associated to the toric Fano 3-fold, while in the other they are non-reduced near the closed point associated to the toric Fano 3-fold. Second, we study K-stability of the general members of two deformation families of smooth Fano 3-folds by building degenerations to K-polystable toric Fano 3-folds.
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Reductivity of the automorphism group of K-polystable Fano varieties

2020-07-23
Jarod Alper , Harold Blum , Daniel Halpern-Leistner +1 more
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K-stability of Fano varieties: an algebro-geometric approach

2020-01-01
Chenyang Xu
We give a survey of the recent progress on the study of K-stability of Fano varieties by an algebro-geometric approach. We give a survey of the recent progress on the study of K-stability of Fano varieties by an algebro-geometric approach.
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On positivity of the CM line bundle on K-moduli spaces

2020-11-01
Chenyang Xu , Ziquan Zhuang
In this paper, we consider the CM line bundle on the K-moduli space, i.e., the moduli space parametrizing K-polystable Fano varieties. We prove it is ample on any proper subspace … In this paper, we consider the CM line bundle on the K-moduli space, i.e., the moduli space parametrizing K-polystable Fano varieties. We prove it is ample on any proper subspace parametrizing reduced uniformly K-stable Fano varieties that conjecturally should be the entire moduli space. As a corollary, we prove that the moduli space parametrizing smoothable K-polystable Fano varieties is projective. During the course of proof, we develop a new invariant for filtrations that can be used to test various K-stability notions of Fano varieties.
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Smoothable del Pezzo surfaces with quotient singularities

2009-12-15
Paul Hacking , Yuri Prokhorov
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.
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Finite generation for valuations computing stability thresholds and applications to K-stability

2022-06-29
Yuchen Liu , Chenyang Xu , Ziquan Zhuang
We prove that on any log Fano pair of dimension $n$ whose stability threshold is less than $\frac{n+1}{n}$, any valuation computing the stability threshold has a finitely generated associated graded … We prove that on any log Fano pair of dimension $n$ whose stability threshold is less than $\frac{n+1}{n}$, any valuation computing the stability threshold has a finitely generated associated graded ring. Together with earlier works, this implies that (a) a log Fano pair is uniformly K-stable (resp. reduced uniformly K-stable) if and only if it is K-stable (resp. K-polystable); (b) the K-moduli spaces are proper and projective; and combining with the previously known equivalence between the existence of Kähler-Einstein metric and reduced uniform K-stability proved by the variational approach, (c) the Yau-Tian-Donaldson conjecture holds for general (possibly singular) log Fano pairs.
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A minimizing valuation is quasi-monomial

2020-04-15
Chenyang Xu
We prove a version of Jonsson-Mustaţǎ's Conjecture, which says for any graded sequence of ideals, there exists a quasi-monomial valuation computing its log canonical threshold. As a corollary, we confirm … We prove a version of Jonsson-Mustaţǎ's Conjecture, which says for any graded sequence of ideals, there exists a quasi-monomial valuation computing its log canonical threshold. As a corollary, we confirm Chi Li's conjecture that a minimizer of the normalized volume function is always quasi-monomial. Applying our techniques to a family of klt singularities, we show that the volume of klt singularities is a constructible function. As a corollary, we prove that in a family of klt log Fano pairs, the K-semistable ones form a Zariski open set. Together with previous works by many people, we conclude that all K-semistable klt Fano varieties with a fixed dimension and volume are parametrized by an Artin stack of finite type, which then admits a separated good moduli space, whose geometric points parametrize K-polystable klt Fano varieties.
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Families of invariant divisors on rational complexity-one T -varieties

2013-07-15
Andreas Hochenegger , Nathan Ilten
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Functors of Artin rings

1968-01-01
Michael Schlessinger
Introduction. In the investigation of functors on the category of preschemes, one is led, by Grothendieck [3], to consider the following situation.Let A be a complete noetherian local ring, u, … Introduction. In the investigation of functors on the category of preschemes, one is led, by Grothendieck [3], to consider the following situation.Let A be a complete noetherian local ring, u, its maximal ideal, and k = A/u.the residue field.(In most applications A is k itself, or a ring of Witt vectors.)Let C be the category of Artin local A-algebras with residue field k.A covariant functor F from C to Sets is called pro-representable if it has the formwhere R is a complete local A-algebra such that R/mn is in C, all n.(m is the maximal ideal in R.)In many cases of interest, F is not pro-representable, but at least one may find an R and a morphism Hom(7?, ■)->■ F of functors such that Hom(.R, A) -> F(A) is surjective for all A in C. If R is chosen suitably "minimal" then R is called a "hull" of F; R is then unique up to noncanonical isomorphism.Theorem 2.11, §2, gives a criterion for F to have a hull, and also a simple criterion for pro-representability which avoids the use of Grothendieck's techniques of nonflat descent [3], in some cases.Grothendieck's program is carried out by Levelt in [4].§3 contains a few geometric applications of these results.To avoid awkward terminology, I have used the word " pro-representable " in a more restrictive sense than Grothendieck [3] has.He considers the category of A-algebras of finite length and allows R to be a projective limit of such rings.The methods of this paper are a simple extension of those used by David Mumford in a proof (unpublished) of the existence of formal moduli for polarized Abelian varieties.I am indebted to Mumford and to John Täte for many valuable suggestions.1.The category CA.Let A be a complete noetherian local ring, with maximal ideal u. and residue field k = A/u.. We define C= CA to be the category of Artinian local A-algebras having residue field k. (That is, the "structure morphism" A -> ,4 of such a ring A induces a trivial extension of residue fields.)Morphisms in C are local homomorphisms of A-algebras.
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An Example of Mirror Symmetry for Fano Threefolds

2020-01-01
Andrea Petracci
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Deformation theory from the point of view of fibered categories

2010-01-01
Mattia Talpo , Angelo Vistoli
We give an exposition of the formal aspects of deformation theory in the language of fibered categories, instead of the more traditional one of functors. The main concepts are that … We give an exposition of the formal aspects of deformation theory in the language of fibered categories, instead of the more traditional one of functors. The main concepts are that of tangent space to a deformation problem, obstruction theory, versal and universal formal deformations. We include proofs of two key results: a versione of Schlessinger's Theorem in this context, and the Ran--Kawamata vanishing theorem for obstructions. We accompany this with a detailed analysis of three important cases: smooth varieties, local complete intersection subschemes and coherent sheaves.

K-stability of Fano varieties: an algebro-geometric approach

2021-08-31
Chenyang Xu
We give a survey of the recent progress on the study of K-stability of Fano varieties by an algebro-geometric approach. We give a survey of the recent progress on the study of K-stability of Fano varieties by an algebro-geometric approach.
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Algebraicity of the metric tangent cones and equivariant K-stability

2021-01-20
Chi Li , Xiaowei Wang , Chenyang Xu
We prove two new results on the <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper K"> <mml:semantics> <mml:mi>K</mml:mi> <mml:annotation encoding="application/x-tex">K</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-polystability of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper Q"> <mml:semantics> <mml:mrow … We prove two new results on the <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper K"> <mml:semantics> <mml:mi>K</mml:mi> <mml:annotation encoding="application/x-tex">K</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-polystability of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper Q"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">Q</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathbb {Q}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-Fano varieties based on purely algebro-geometric arguments. The first one says that any <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper K"> <mml:semantics> <mml:mi>K</mml:mi> <mml:annotation encoding="application/x-tex">K</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-semistable log Fano cone has a special degeneration to a uniquely determined <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper K"> <mml:semantics> <mml:mi>K</mml:mi> <mml:annotation encoding="application/x-tex">K</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-polystable log Fano cone. As a corollary, we combine it with the differential-geometric results to complete the proof of Donaldson-Sun’s conjecture which says that the metric tangent cone of any point appearing on a Gromov-Hausdorff limit of Kähler-Einstein Fano manifolds depends only on the algebraic structure of the singularity. The second result says that for any log Fano variety with the torus action, <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper K"> <mml:semantics> <mml:mi>K</mml:mi> <mml:annotation encoding="application/x-tex">K</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-polystability is equivalent to equivariant <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper K"> <mml:semantics> <mml:mi>K</mml:mi> <mml:annotation encoding="application/x-tex">K</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-polystability, that is, to check <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper K"> <mml:semantics> <mml:mi>K</mml:mi> <mml:annotation encoding="application/x-tex">K</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-polystability, it is sufficient to check special test configurations which are equivariant under the torus action.
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From real affine geometry to complex geometry

2011-10-14
Mark Gross , Bernd Siebert
We construct from a real affine manifold with singularities (a tropical manifold) a degeneration of Calabi-Yau manifolds.This solves a fundamental problem in mirror symmetry.Furthermore, a striking feature of our approach … We construct from a real affine manifold with singularities (a tropical manifold) a degeneration of Calabi-Yau manifolds.This solves a fundamental problem in mirror symmetry.Furthermore, a striking feature of our approach is that it yields an explicit and canonical order-by-order description of the degeneration via families of tropical trees.This gives complete control of the B-model side of mirror symmetry in terms of tropical geometry.For example, we expect that our deformation parameter is a canonical coordinate, and expect period calculations to be expressible in terms of tropical curves.We anticipate this will lead to a proof of mirror symmetry via tropical methods.
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Cracked polytopes and Fano toric complete intersections

2019-09-16
Thomas Prince
Abstract We introduce the notion of cracked polytope , and – making use of joint work with Coates and Kasprzyk—construct the associated toric variety X as a subvariety of a … Abstract We introduce the notion of cracked polytope , and – making use of joint work with Coates and Kasprzyk—construct the associated toric variety X as a subvariety of a smooth toric variety Y under certain conditions. Restricting to the case in which this subvariety is a complete intersection, we present a sufficient condition for a smoothing of X to exist inside Y . We exhibit a relative anti-canonical divisor for this smoothing of X , and show that the general member is simple normal crossings.
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From cracked polytopes to Fano threefolds

2020-02-22
Thomas Prince
Abstract We construct Fano threefolds with very ample anti-canonical bundle and Picard rank greater than one from cracked polytopes —polytopes whose intersection with a complete fan forms a set of … Abstract We construct Fano threefolds with very ample anti-canonical bundle and Picard rank greater than one from cracked polytopes —polytopes whose intersection with a complete fan forms a set of unimodular polytopes—using Laurent inversion; a method developed jointly with Coates–Kasprzyk. We also give constructions of rank one Fano threefolds from cracked polytopes, following work of Christophersen–Ilten and Galkin. We explore the problem of classifying polytopes cracked along a given fan in three dimensions, and classify the unimodular polytopes which can occur as ‘pieces’ of a cracked polytope.
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Period integrals from wall structures via tropical cycles, canonical coordinates in mirror symmetry and analyticity of toric degenerations

2020-05-14
Helge Ruddat , Bernd Siebert
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Mirror Symmetry and Smoothing Gorenstein Toric Affine 3-folds

2022-03-14
Alessio Corti , Matej Filip , Andrea Petracci
A summary is not available for this content so a preview has been provided. Please use the Get access link above for information on how to access this content. A summary is not available for this content so a preview has been provided. Please use the Get access link above for information on how to access this content.
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Terminal Fano threefolds and their smoothings

2010-09-24
Priska Jahnke , Ivo Radloff
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On the quantum periods of del Pezzo surfaces with ⅓ (1, 1) singularities

2018-04-06
Alessandro Oneto , Andrea Petracci
Abstract In earlier joint work with collaborators we gave a conjectural classification of a broad class of orbifold del Pezzo surfaces, using Mirror Symmetry. We proposed that del Pezzo surfaces … Abstract In earlier joint work with collaborators we gave a conjectural classification of a broad class of orbifold del Pezzo surfaces, using Mirror Symmetry. We proposed that del Pezzo surfaces X with isolated cyclic quotient singularities such that X admits a ℚ-Gorenstein toric degeneration correspond via Mirror Symmetry to maximally mutable Laurent polynomials f in two variables, and that the quantum period of such a surface X , which is a generating function for Gromov–Witten invariants of X , coincides with the classical period of its mirror partner f . In this paper we give strong evidence for this conjecture. Contingent on conjectural generalisations of the Quantum Lefschetz theorem and the Abelian/non-Abelian correspondence, we compute many quantum periods for del Pezzo surfaces with <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"><m:mtable><m:mtr><m:mtd><m:mfrac><m:mn>1</m:mn><m:mn>3</m:mn></m:mfrac></m:mtd></m:mtr></m:mtable></m:math> $\begin{array}{} \frac{1}{3} \end{array} $ (1, 1) singularities. Our computations also give strong evidence for the extension of these two principles to the orbifold setting.
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Hodge theoretic aspects of mirror symmetry

2008-01-01
Ludmil Katzarkov , Maxim Kontsevich , Tony Pantev
We discuss the Hodge theory of algebraic non-commutative spaces and analyze how this theory interacts with the Calabi-Yau condition and with mirror symmetry. We develop an abstract theory of non-commutative … We discuss the Hodge theory of algebraic non-commutative spaces and analyze how this theory interacts with the Calabi-Yau condition and with mirror symmetry. We develop an abstract theory of non-commutative Hodge structures, investigate existence and variations, and propose explicit construction and classification techniques. We study the main examples of non-commutative Hodge structures coming from a symplectic or a complex geometry possibly twisted by a potential. We study the interactions of the new Hodge theoretic invariants with mirror symmetry and derive non-commutative analogues of some of the more interesting consequences of Hodge theory such as unobstructedness and the construction of canonical coordinates on moduli.
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Computation of the vector space T1 for affine toric varieties

1994-08-01
Klaus Altmann

On the abstract Bogomolov-Tian-Todorov Theorem

2017-06-01
Donatella Iacono
We describe an abstract version of the Theorem of Bogomolov-Tian-Todorov, whose underlying idea is already contained in various papers by Bandiera, Fiorenza, Iacono, Manetti. More explicitly, we prove an algebraic … We describe an abstract version of the Theorem of Bogomolov-Tian-Todorov, whose underlying idea is already contained in various papers by Bandiera, Fiorenza, Iacono, Manetti. More explicitly, we prove an algebraic criterion for a differential graded Lie algebras to be homotopy abelian. Then, we collect together many examples and applications in deformation theory and other settings.
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Some examples of non-smoothable Gorenstein Fano toric threefolds

2019-08-03
Andrea Petracci
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On Landau–Ginzburg models for Fano varieties

2007-01-01
Victor Przyjalkowski
On Landau-Ginzburg models for Fano varieties On Landau-Ginzburg models for Fano varieties
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Geometry of logarithmic forms and deformations of complex structures

2019-07-19
Kefeng Liu , Sheng Rao , Xueyuan Wan
We present a new method to solve certain <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="ModifyingAbove partial-differential With bar"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mover> <mml:mi mathvariant="normal">∂<!-- ∂ --></mml:mi> <mml:mo stretchy="false">¯<!-- ¯ --></mml:mo> </mml:mover> </mml:mrow> … We present a new method to solve certain <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="ModifyingAbove partial-differential With bar"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mover> <mml:mi mathvariant="normal">∂<!-- ∂ --></mml:mi> <mml:mo stretchy="false">¯<!-- ¯ --></mml:mo> </mml:mover> </mml:mrow> <mml:annotation encoding="application/x-tex">\bar \partial</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-equations for logarithmic differential forms by using harmonic integral theory for currents on Kähler manifolds. The result can be considered as a <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="partial-differential ModifyingAbove partial-differential With bar"> <mml:semantics> <mml:mrow> <mml:mi mathvariant="normal">∂<!-- ∂ --></mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mover> <mml:mi mathvariant="normal">∂<!-- ∂ --></mml:mi> <mml:mo stretchy="false">¯<!-- ¯ --></mml:mo> </mml:mover> </mml:mrow> </mml:mrow> <mml:annotation encoding="application/x-tex">\partial \bar \partial</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-lemma for logarithmic forms. As applications, we generalize the result of Deligne about closedness of logarithmic forms, give geometric and simpler proofs of Deligne’s degeneracy theorem for the logarithmic Hodge to de Rham spectral sequences at <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper E 1"> <mml:semantics> <mml:msub> <mml:mi>E</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:annotation encoding="application/x-tex">E_1</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-level, as well as a certain injectivity theorem on compact Kähler manifolds. Furthermore, for a family of logarithmic deformations of complex structures on Kähler manifolds, we construct the extension for any logarithmic <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis n comma q right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>n</mml:mi> <mml:mo>,</mml:mo> <mml:mi>q</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">(n,q)</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-form on the central fiber and thus deduce the local stability of log Calabi-Yau structure by extending an iteration method to the logarithmic forms. Finally we prove the unobstructedness of the deformations of a log Calabi-Yau pair and a pair on a Calabi-Yau manifold by the differential geometric method.
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Algebraicity of the Metric Tangent Cones and Equivariant K-stability

2018-01-01
Chi Li , Xiaowei Wang , Chenyang Xu
We prove two new results on the K-polystability of Q-Fano varieties based on purely algebro-geometric arguments. The first one says that any K-semistable log Fano cone has a special degeneration … We prove two new results on the K-polystability of Q-Fano varieties based on purely algebro-geometric arguments. The first one says that any K-semistable log Fano cone has a special degeneration to a uniquely determined K-polystable log Fano cone. As a corollary, we combine it with the differential-geometric results to complete the proof of Donaldson-Sun's Conjecture which says that the metric tangent cone of any close point appearing on a Gromov-Hausdorff limit of Kahler-Einstein Fano manifolds only depends on the algebraic structure of the singularity. The second result says that for any log Fano variety with a torus action, the K-polystability is equivalent to the equivariant K-polystability, that is, to check K-polystability, it is sufficient to check special test configurations which are equivariant under the torus action.
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An algebraic proof of Bogomolov-Tian-Todorov theorem

2010-01-01
Donatella Iacono , Marco Manetti
We give a completely algebraic proof of the Bogomolov-Tian-Todorov theorem. More precisely, we shall prove that if X is a smooth projective variety with trivial canonical bundle defined over an … We give a completely algebraic proof of the Bogomolov-Tian-Todorov theorem. More precisely, we shall prove that if X is a smooth projective variety with trivial canonical bundle defined over an algebraically closed field of characteristic 0, then the L-infinity algebra governing infinitesimal deformations of X is quasi-isomorphic to an abelian differential graded Lie algebra.
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A New Cohomology Theory of Orbifold

2004-05-04
Chen Weimin , Yongbin Ruan
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