We compute global log canonical thresholds of some smooth Fano threefolds.Contents 1. Introduction 2. Preliminaries 3. The Mukai-Umemura threefold 4. Cubic surfaces 5. Del Pezzo surfaces 6. Toric varieties 7. …
We compute global log canonical thresholds of some smooth Fano threefolds.Contents 1. Introduction 2. Preliminaries 3. The Mukai-Umemura threefold 4. Cubic surfaces 5. Del Pezzo surfaces 6. Toric varieties 7. Del Pezzo threefolds 8. Fano threefolds with ρ = 2 9. Fano threefolds with ρ = 3 10.Fano threefolds with ρ 4 11.Upper bounds Appendix A. By Jean-Pierre Demailly.On Tian's invariant and log canonical thresholds Appendix B. The Big Table References U α ψ
The birational superrigidity and, in particular, the non-rationality of a smooth three-dimensional quartic was proved by V. Iskovskikh and Yu. Manin in 1971, and this led immediately to a counterexample …
The birational superrigidity and, in particular, the non-rationality of a smooth three-dimensional quartic was proved by V. Iskovskikh and Yu. Manin in 1971, and this led immediately to a counterexample to the three-dimensional Luroth problem. Since then, birational rigidity and superrigidity have been proved for a broad class of higher-dimensional varieties, among which the Fano varieties occupy the central place. The present paper is a survey of the theory of birationally rigid Fano varieties.
We show that affine cones over smooth cubic surfaces do not admit non-trivial \mathbb{G}_{a} -actions.
We show that affine cones over smooth cubic surfaces do not admit non-trivial \mathbb{G}_{a} -actions.
We classify Fano 3-folds with canonical Gorenstein singularities whose anticanonical linear system has no base points but does not give an embedding, and we classify anticanonically embedded Fano 3-folds with …
We classify Fano 3-folds with canonical Gorenstein singularities whose anticanonical linear system has no base points but does not give an embedding, and we classify anticanonically embedded Fano 3-folds with canonical Gorenstein singu- larities which are not intersections of quadrics. We also study rationality questions for most of these varieties.
A lower bound for global log canonical thresholds on smooth hypersurfaces is found. This bound cannot be improved for the fixed degree and dimension of the hypersurface.
A lower bound for global log canonical thresholds on smooth hypersurfaces is found. This bound cannot be improved for the fixed degree and dimension of the hypersurface.
It is known that the -ޑfactoriality of a nodal quartic 3-fold in ސ 4 implies its nonrationality.We prove that a nodal quartic 3-fold with at most 8 nodes is -ޑfactorial, …
It is known that the -ޑfactoriality of a nodal quartic 3-fold in ސ 4 implies its nonrationality.We prove that a nodal quartic 3-fold with at most 8 nodes is -ޑfactorial, while one with 9 nodes is not -ޑfactorial if and only if it contains a plane.There are nonrational non--ޑfactorial nodal quartic 3-folds.In particular, we prove the nonrationality of a general non--ޑfactorial nodal quartic 3-fold that contains either a plane or a smooth del Pezzo surface of degree 4.
We propose a new method to study birational maps between Fano varieties based on multiplier ideal sheaves. Using this method, we prove equivariant birational rigidity of four Fano threefolds acted …
We propose a new method to study birational maps between Fano varieties based on multiplier ideal sheaves. Using this method, we prove equivariant birational rigidity of four Fano threefolds acted on by the group $\mathrm {A}_6$. As an application, we obtain that $\mathrm {Bir}(\mathbb {P}^{3})$ has at least five non-conjugate subgroups isomorphic to $\mathrm {A}_{6}$.
We study exceptional quotient singularities.In particular, we prove an exceptionality criterion in terms of the ˛-invariant of Tian, and utilize it to classify four-dimensional and five-dimensional exceptional quotient singularities.We assume …
We study exceptional quotient singularities.In particular, we prove an exceptionality criterion in terms of the ˛-invariant of Tian, and utilize it to classify four-dimensional and five-dimensional exceptional quotient singularities.We assume that all varieties are projective, normal, and defined over C .
Algebraic varieties are shapes defined by polynomial equations. Smooth Fano threefolds are a fundamental subclass that can be thought of as higher-dimensional generalizations of ordinary spheres. They belong to 105 …
Algebraic varieties are shapes defined by polynomial equations. Smooth Fano threefolds are a fundamental subclass that can be thought of as higher-dimensional generalizations of ordinary spheres. They belong to 105 irreducible deformation families. This book determines whether the general element of each family admits a Kähler–Einstein metric (and for many families, for all elements), addressing a question going back to Calabi 70 years ago. The book's solution exploits the relation between these metrics and the algebraic notion of K-stability. Moreover, the book presents many different techniques to prove the existence of a Kähler–Einstein metric, containing many additional relevant results such as the classification of all Kähler–Einstein smooth Fano threefolds with infinite automorphism groups and computations of delta-invariants of all smooth del Pezzo surfaces. This book will be essential reading for researchers and graduate students working on algebraic geometry and complex geometry.
We prove the $\mathbb {Q}$-factoriality of a nodal hypersurface in $\mathbb {P}^{4}$ of degree $n$ with at most ${\frac {(n-1)^{2}}{4}}$ nodes and the $\mathbb {Q}$-factoriality of a double cover of …
We prove the $\mathbb {Q}$-factoriality of a nodal hypersurface in $\mathbb {P}^{4}$ of degree $n$ with at most ${\frac {(n-1)^{2}}{4}}$ nodes and the $\mathbb {Q}$-factoriality of a double cover of $\mathbb {P}^{3}$ branched over a nodal surface of degree $2r$ with at most ${\frac {(2r-1)r}{3}}$ nodes.
We study the rationality problem for nodal quartic double solids. In particular, we prove that nodal quartic double solids with at most six singular points are irrational, and nodal quartic …
We study the rationality problem for nodal quartic double solids. In particular, we prove that nodal quartic double solids with at most six singular points are irrational, and nodal quartic double solids with at least eleven singular points are rational.
Let X be a smooth hypersurface of degree n 3i nP n .I t is proved that the log canonical threshold of an arbitrary hyperplane section H of it is …
Let X be a smooth hypersurface of degree n 3i nP n .I t is proved that the log canonical threshold of an arbitrary hyperplane section H of it is at least (n − 1)/n. Under the assumption of the log minimal model program it is also proved that the log canonical threshold of H ⊂ X is (n − 1)/n if and only if H is a cone in P n−1 over a smooth hypersurface of degree n in P n−2 . Bibliography: 16 titles.
We construct an example of the birationally rigid complete intersection of a quadric and a cubic in $\PA^5$ with an ordinary double point, which under a small deformation gives a …
We construct an example of the birationally rigid complete intersection of a quadric and a cubic in $\PA^5$ with an ordinary double point, which under a small deformation gives a non-rigid Fano variety. Thus we show that birational rigidity is not open in moduli.
The non-rationality and the birational superrigidity is proved for the 4-dimensional smooth complete intersection of a quadric and a quartic in that contains no 2-dimensional linear subspace of . It …
The non-rationality and the birational superrigidity is proved for the 4-dimensional smooth complete intersection of a quadric and a quartic in that contains no 2-dimensional linear subspace of . It is also proved that such an intersection is not birationally isomorphic to an elliptic fibration.
We study del Pezzo surfaces that are quasismooth and well-formed weighted hypersurfaces. In particular, we find all such surfaces whose α-invariant of Tian is greater than 2/3.
We study del Pezzo surfaces that are quasismooth and well-formed weighted hypersurfaces. In particular, we find all such surfaces whose α-invariant of Tian is greater than 2/3.
We prove that every quasi-smooth weighted Fano threefold hypersurface in the 95 families of Fletcher and Reid is birationally rigid.
We prove that every quasi-smooth weighted Fano threefold hypersurface in the 95 families of Fletcher and Reid is birationally rigid.
76 В. В. Пржиялковский, И. А. Чельцов, К. А. Шрамов Гиперэллиптические и тригональные трехмерные многообразия Фано Классифицированы трехмерные многообразия Фано с каноническими горенштейновыми особенностями, антиканоническая линейная система которых не имеет …
76 В. В. Пржиялковский, И. А. Чельцов, К. А. Шрамов Гиперэллиптические и тригональные трехмерные многообразия Фано Классифицированы трехмерные многообразия Фано с каноническими горенштейновыми особенностями, антиканоническая линейная система которых не имеет базисных точек, но не задает вложения.Также классифицированы антиканоническивложенныетрехмерныемногообразия Фано с каноническими горенштейновыми особенностями, которые не являются пересечением квадрик.Для большинства полученных многообразий доказываются утверждения об их ра циональности или
We prove the factoriality of a nodal hypersurface in P 4 of degree d that has at most 2(d -1) 2 /3 singular points, and we prove the factoriality of …
We prove the factoriality of a nodal hypersurface in P 4 of degree d that has at most 2(d -1) 2 /3 singular points, and we prove the factoriality of a double cover of P 3 branched over a nodal surface of degree 2r having less than (2r -1)r singular points.
The birational geometry of an arbitrary smooth quintic 4-fold is studied using the properties of log pairs. As a result, a new proof of its birational rigidity is given and …
The birational geometry of an arbitrary smooth quintic 4-fold is studied using the properties of log pairs. As a result, a new proof of its birational rigidity is given and all birational maps of a smooth quintic 4-fold into fibrations with general fibre of Kodaira dimension zero are described.In the Addendum similar results are obtained for all smooth hypersurfaces of degree in in the case of equal to 6, 7, or 8.
For each del Pezzo surface $S$ with du Val singularities, we determine whether it admits a $(-K_S)$-polar cylinder or not. If it allows one, then we present an effective $\mathbb{Q}$-divisor …
For each del Pezzo surface $S$ with du Val singularities, we determine whether it admits a $(-K_S)$-polar cylinder or not. If it allows one, then we present an effective $\mathbb{Q}$-divisor $D$ that is $\mathbb{Q}$-linearly equivalent to $-K_S$ and such that the open set $S\setminus\mathrm{Supp}(D)$ is a cylinder. As a corollary, we classify all the del Pezzo surfaces with du Val singularities that admit nontrivial $\mathbb{G}_a$-actions on their affine cones defined by their anticanonical divisors.
We study birational transformations into elliptic fibrations and birational automorphisms of quasismooth anticanonically embedded weighted Fano 3-fold hypersurfaces with terminal singularities classified by A. R. Iano-Fletcher, J. Johnson, J. Kollár, …
We study birational transformations into elliptic fibrations and birational automorphisms of quasismooth anticanonically embedded weighted Fano 3-fold hypersurfaces with terminal singularities classified by A. R. Iano-Fletcher, J. Johnson, J. Kollár, and M. Reid.
We prove two new local inequalities for divisors on smooth surfaces and consider several applications of these inequalities.
We prove two new local inequalities for divisors on smooth surfaces and consider several applications of these inequalities.
On del Pezzo surfaces, we study effective ample |$\mathbb{R}$|-divisors such that the complements of their supports are isomorphic to |$\mathbb{A}^1$|-bundles over smooth affine curves. All considered varieties are assumed to …
On del Pezzo surfaces, we study effective ample |$\mathbb{R}$|-divisors such that the complements of their supports are isomorphic to |$\mathbb{A}^1$|-bundles over smooth affine curves. All considered varieties are assumed to be algebraic and defined over an algebraically closed field of characteristic |$0$| throughout this article.
Abstract. We show that infinitely many Gorenstein weakly-exceptional quotient singularities exist in all dimensions, we prove a weak-exceptionality criterion for five-dimensional quotient singularities, and we find a sufficient condition for …
Abstract. We show that infinitely many Gorenstein weakly-exceptional quotient singularities exist in all dimensions, we prove a weak-exceptionality criterion for five-dimensional quotient singularities, and we find a sufficient condition for being weakly-exceptional for six-dimensional quotient singularities. The proof is naturally linked to various classical geometrical constructions related to subvarieties of small degree in projective spaces, in particular Bordiga surfaces and Bordiga threefolds.
We prove that $$\delta $$ -invariants of smooth cubic surfaces are at least $$\frac{6}{5}$$ .
We prove that $$\delta $$ -invariants of smooth cubic surfaces are at least $$\frac{6}{5}$$ .
Abstract We study linearizability of actions of finite groups on singular cubic threefolds, using cohomological tools, intermediate Jacobians, Burnside invariants, and the equivariant Minimal Model Program.
Abstract We study linearizability of actions of finite groups on singular cubic threefolds, using cohomological tools, intermediate Jacobians, Burnside invariants, and the equivariant Minimal Model Program.
We find all K-polystable limits of divisors in $(\mathbb{P}^1)^4$ of degree $(1,1,1,1)$ and explicitly describe the associated irreducible component of the K-moduli space.
We find all K-polystable limits of divisors in $(\mathbb{P}^1)^4$ of degree $(1,1,1,1)$ and explicitly describe the associated irreducible component of the K-moduli space.
Abstract We study linearizability and stable linearizability of actions of finite groups on the Segre cubic and Burkhardt quartic, using techniques from group cohomology, birational rigidity, and the Burnside formalism.
Abstract We study linearizability and stable linearizability of actions of finite groups on the Segre cubic and Burkhardt quartic, using techniques from group cohomology, birational rigidity, and the Burnside formalism.
Priors with non-smooth log densities have been widely used in Bayesian inverse problems, particularly in imaging, due to their sparsity inducing properties. To date, the majority of algorithms for handling …
Priors with non-smooth log densities have been widely used in Bayesian inverse problems, particularly in imaging, due to their sparsity inducing properties. To date, the majority of algorithms for handling such densities are based on proximal Langevin dynamics where one replaces the non-smooth part by a smooth approximation known as the Moreau envelope. In this work, we introduce a novel approach for sampling densities with $\ell_1$-priors based on a Hadamard product parameterization. This builds upon the idea that the Laplace prior has a Gaussian mixture representation and our method can be seen as a form of overparametrization: by increasing the number of variables, we construct a density from which one can directly recover the original density. This is fundamentally different from proximal-type approaches since our resolution is exact, while proximal-based methods introduce additional bias due to the Moreau-envelope smoothing. For our new density, we present its Langevin dynamics in continuous time and establish well-posedness and geometric ergodicity. We also present a discretization scheme for the continuous dynamics and prove convergence as the time-step diminishes.
We explore connections between existence of $\Bbbk$-rational points for Fano varieties defined over $\Bbbk$, a subfield of $\mathbb{C}$, and existence of K\"ahler-Einstein metrics on their geometric models. First, we show …
We explore connections between existence of $\Bbbk$-rational points for Fano varieties defined over $\Bbbk$, a subfield of $\mathbb{C}$, and existence of K\"ahler-Einstein metrics on their geometric models. First, we show that geometric models of del Pezzo surfaces with at worst quotient singularities defined over $\Bbbk\subset\mathbb{C}$ admit (orbifold) K\"ahler--Einstein metrics if they do not have $\Bbbk$-rational points. Then we prove the same result for smooth Fano 3-folds with 8 exceptions. Consequently, we explicitly describe several families of pointless Fano 3-folds whose geometric models admit K\"ahler-Einstein metrics. In particular, we obtain new examples of prime Fano 3-folds of genus $12$ that admit K\"ahler--Einstein metrics. Our result can also be used to prove existence of rational points for certain Fano varieties, for example for any smooth Fano 3-fold over $\Bbbk\subset\mathbb{C}$ whose geometric model is strictly K-semistable.
Abstract We introduce a new subclass of Fano varieties (Casagrande–Druel varieties) that are 𝑛-dimensional varieties constructed from Fano double covers of dimension <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>n</m:mi> <m:mo>−</m:mo> <m:mn>1</m:mn> </m:mrow> </m:math> …
Abstract We introduce a new subclass of Fano varieties (Casagrande–Druel varieties) that are 𝑛-dimensional varieties constructed from Fano double covers of dimension <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>n</m:mi> <m:mo>−</m:mo> <m:mn>1</m:mn> </m:mrow> </m:math> n-1 . We conjecture that a Casagrande–Druel variety is K-polystable if the double cover and its base space are K-polystable. We prove this for smoothable Casagrande–Druel threefolds, and for Casagrande–Druel varieties constructed from double covers of <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mi mathvariant="double-struck">P</m:mi> <m:mrow> <m:mi>n</m:mi> <m:mo>−</m:mo> <m:mn>1</m:mn> </m:mrow> </m:msup> </m:math> \mathbb{P}^{n-1} ramified over smooth hypersurfaces of degree <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mn>2</m:mn> <m:mo></m:mo> <m:mi>d</m:mi> </m:mrow> </m:math> 2d with <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>n</m:mi> <m:mo>></m:mo> <m:mi>d</m:mi> <m:mo>></m:mo> <m:mfrac> <m:mi>n</m:mi> <m:mn>2</m:mn> </m:mfrac> <m:mo>></m:mo> <m:mn>1</m:mn> </m:mrow> </m:math> n>d>\frac{n}{2}>1 . As an application, we describe the connected components of the K-moduli space parametrizing smoothable K-polystable Fano threefolds in the families № 3.9 and № 4.2 in the Mori–Mukai classification.
By identifying K-polystable limits in 4 specific deformations families of smooth Fano 3-folds, we complete the classification of one-dimensional components in the K-moduli space of smoothable Fano 3-folds.
By identifying K-polystable limits in 4 specific deformations families of smooth Fano 3-folds, we complete the classification of one-dimensional components in the K-moduli space of smoothable Fano 3-folds.
We classify birational involutions of the real projective plane up to conjugation. In contrast with an analogous classification over the complex numbers (due to E. Bertini, G. Castelnuovo, F. Enriques, …
We classify birational involutions of the real projective plane up to conjugation. In contrast with an analogous classification over the complex numbers (due to E. Bertini, G. Castelnuovo, F. Enriques, L. Bayle and A. Beauville), which includes four different classes of involutions, we discover 12 different classes over the reals, and provide many examples when the fixed curve of an involution does not determine its conjugacy class in the real plane Cremona group.
We study nodal del Pezzo 3 -folds of degree 1 (also known as double Veronese cones) with 28 singularities, which is the maximal possible number of singularities for such varieties. …
We study nodal del Pezzo 3 -folds of degree 1 (also known as double Veronese cones) with 28 singularities, which is the maximal possible number of singularities for such varieties. We show that they are in one-to-one correspondence with smooth plane quartics and use this correspondence to study their automorphism groups. As an application, we find all G -birationally rigid varieties of this kind, and construct an infinite number of non-conjugate embeddings of the group \mathfrak{S}_{4} into the space Cremona group.
We classify non-factorial nodal Fano threefolds with 1 node and class group of rank 2 .
We classify non-factorial nodal Fano threefolds with 1 node and class group of rank 2 .
Abstract In this note, we study K-stability of smooth Fano threefolds that can be obtained by blowing up the three-dimensional projective space along a smooth elliptic curve of degree five.
Abstract In this note, we study K-stability of smooth Fano threefolds that can be obtained by blowing up the three-dimensional projective space along a smooth elliptic curve of degree five.
We study linearizability of actions of finite groups on cubic threefolds with nonnodal isolated singularities.
We study linearizability of actions of finite groups on cubic threefolds with nonnodal isolated singularities.
We study Fano threefolds that can be obtained by blowing up the three-dimensional projective space along a smooth curve of degree six and genus three. We produce many new K-stable …
We study Fano threefolds that can be obtained by blowing up the three-dimensional projective space along a smooth curve of degree six and genus three. We produce many new K-stable examples of such threefolds, and we describe all finite groups that can act faithfully on them.
We prove K-stability of smooth Fano 3-folds of Picard rank 3 and degree 22 that satisfy very explicit generality condition.
We prove K-stability of smooth Fano 3-folds of Picard rank 3 and degree 22 that satisfy very explicit generality condition.
We study linearizability of actions of finite groups on singular cubic threefolds, using cohomological tools, intermediate Jacobians, Burnside invariants, and the equivariant Minimal Model Program.
We study linearizability of actions of finite groups on singular cubic threefolds, using cohomological tools, intermediate Jacobians, Burnside invariants, and the equivariant Minimal Model Program.
Abstract We prove that all smooth Fano threefolds in the families and are K-stable, and we also prove that smooth Fano threefolds in the family that satisfy one very explicit …
Abstract We prove that all smooth Fano threefolds in the families and are K-stable, and we also prove that smooth Fano threefolds in the family that satisfy one very explicit generality condition are K-stable.
Algebraic varieties are shapes defined by polynomial equations. Smooth Fano threefolds are a fundamental subclass that can be thought of as higher-dimensional generalizations of ordinary spheres. They belong to 105 …
Algebraic varieties are shapes defined by polynomial equations. Smooth Fano threefolds are a fundamental subclass that can be thought of as higher-dimensional generalizations of ordinary spheres. They belong to 105 irreducible deformation families. This book determines whether the general element of each family admits a Kähler–Einstein metric (and for many families, for all elements), addressing a question going back to Calabi 70 years ago. The book's solution exploits the relation between these metrics and the algebraic notion of K-stability. Moreover, the book presents many different techniques to prove the existence of a Kähler–Einstein metric, containing many additional relevant results such as the classification of all Kähler–Einstein smooth Fano threefolds with infinite automorphism groups and computations of delta-invariants of all smooth del Pezzo surfaces. This book will be essential reading for researchers and graduate students working on algebraic geometry and complex geometry.
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Abstract We prove that every smooth divisor in $\mathbb {P}^1\times \mathbb {P}^1\times \mathbb {P}^2$ of degree $(1,1,2)$ is K-stable.
Abstract We prove that every smooth divisor in $\mathbb {P}^1\times \mathbb {P}^1\times \mathbb {P}^2$ of degree $(1,1,2)$ is K-stable.
We study the problem of existence of Kähler–Einstein metrics on smooth Fano threefolds of Picard rank one and anticanonical degree <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="22"> <mml:semantics> <mml:mn>22</mml:mn> <mml:annotation encoding="application/x-tex">22</mml:annotation> </mml:semantics> …
We study the problem of existence of Kähler–Einstein metrics on smooth Fano threefolds of Picard rank one and anticanonical degree <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="22"> <mml:semantics> <mml:mn>22</mml:mn> <mml:annotation encoding="application/x-tex">22</mml:annotation> </mml:semantics> </mml:math> </inline-formula> that admit a faithful action of the multiplicative group <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper C Superscript asterisk"> <mml:semantics> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">C</mml:mi> </mml:mrow> <mml:mo>∗<!-- ∗ --></mml:mo> </mml:msup> <mml:annotation encoding="application/x-tex">\mathbb {C}^\ast</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. We prove that, with the possible exception of two explicitly described cases, all such smooth Fano threefolds are Kähler–Einstein.
Abstract We study toric G -solid Fano threefolds that have at most terminal singularities, where G is an algebraic subgroup of the normalizer of a maximal torus in their automorphism …
Abstract We study toric G -solid Fano threefolds that have at most terminal singularities, where G is an algebraic subgroup of the normalizer of a maximal torus in their automorphism groups. All varieties are assumed to be projective and defined over the field of complex numbers.
We prove that for every $\epsilon>0$, there is a birationally super-rigid Fano variety $X$ such that $\frac{1}{2}\leqslant\alpha(X)\leqslant \frac{1}{2}+\epsilon$. Also we show that for every $\epsilon>0$, there is a Fano variety …
We prove that for every $\epsilon>0$, there is a birationally super-rigid Fano variety $X$ such that $\frac{1}{2}\leqslant\alpha(X)\leqslant \frac{1}{2}+\epsilon$. Also we show that for every $\epsilon>0$, there is a Fano variety $X$ and a finite subgroup $G\subset\mathrm{Aut}(X)$ such that $X$ is $G$-birationally super-rigid, and $\alpha_G(X)<\epsilon$.
We prove that smooth Fano 3-folds in the families 2.18 and 3.4 are K-stable.
We prove that smooth Fano 3-folds in the families 2.18 and 3.4 are K-stable.
We classify non-factorial nodal Fano threefolds with $1$ node and class group of rank $2$.
We classify non-factorial nodal Fano threefolds with $1$ node and class group of rank $2$.
We study linearizability and stable linearizability of actions of finite groups on the Segre cubic and Burkhardt quartic, using techniques from group cohomology, birational rigidity, and the Burnside formalism.
We study linearizability and stable linearizability of actions of finite groups on the Segre cubic and Burkhardt quartic, using techniques from group cohomology, birational rigidity, and the Burnside formalism.
We introduce a new subclass of Fano varieties (Casagrande-Druel varieties), that are $n$-dimensional varieties constructed from Fano double covers of dimension $n-1$. We conjecture that a Casagrande-Druel variety is K-polystable …
We introduce a new subclass of Fano varieties (Casagrande-Druel varieties), that are $n$-dimensional varieties constructed from Fano double covers of dimension $n-1$. We conjecture that a Casagrande-Druel variety is K-polystable if the double cover and its base space are K-polystable. We prove this for smoothable Casagrande-Druel threefolds, and for Casagrande-Druel varieties constructed from double covers of $\mathbb{P}^{n-1}$ ramified over smooth hypersurfaces of degree $2d$ with $n>d>\frac{n}{2}>1$. As an application, we describe the connected components of the K-moduli space parametrizing smoothable K-polystable Fano threefolds in the families 3.9 and 4.2 in the Mori-Mukai classification.
By identifying K-polystable limits in 4 specific deformations families of smooth Fano 3-folds, we complete the classification of one-dimensional components in the K-moduli space of smoothable Fano 3-folds.
By identifying K-polystable limits in 4 specific deformations families of smooth Fano 3-folds, we complete the classification of one-dimensional components in the K-moduli space of smoothable Fano 3-folds.
In this note, we study K-stability of smooth Fano threefolds that can be obtained by blowing up the three-dimensional projective space along a smooth elliptic curve of degree five.
In this note, we study K-stability of smooth Fano threefolds that can be obtained by blowing up the three-dimensional projective space along a smooth elliptic curve of degree five.
Abstract We show that the only finite quasi-simple non-abelian groups that can faithfully act on rationally connected threefolds are the following groups: ${\mathfrak{A}}_5$ , ${\text{PSL}}_2(\textbf{F}_7)$ , ${\mathfrak{A}}_6$ , ${\text{SL}}_2(\textbf{F}_8)$ , …
Abstract We show that the only finite quasi-simple non-abelian groups that can faithfully act on rationally connected threefolds are the following groups: ${\mathfrak{A}}_5$ , ${\text{PSL}}_2(\textbf{F}_7)$ , ${\mathfrak{A}}_6$ , ${\text{SL}}_2(\textbf{F}_8)$ , ${\mathfrak{A}}_7$ , ${\text{PSp}}_4(\textbf{F}_3)$ , ${\text{SL}}_2(\textbf{F}_{7})$ , $2.{\mathfrak{A}}_5$ , $2.{\mathfrak{A}}_6$ , $3.{\mathfrak{A}}_6$ or $6.{\mathfrak{A}}_6$ . All of these groups with a possible exception of $2.{\mathfrak{A}}_6$ and $6.{\mathfrak{A}}_6$ indeed act on some rationally connected threefolds.
Abstract We find all K-stable smooth Fano threefolds in the family No. 2.22.
Abstract We find all K-stable smooth Fano threefolds in the family No. 2.22.
Abstract We study equivariant birational geometry of (rational) quartic double solids ramified over (singular) Kummer surfaces.
Abstract We study equivariant birational geometry of (rational) quartic double solids ramified over (singular) Kummer surfaces.
We study the complexity of birational self-maps of a projective threefold $X$ by looking at the birational type of surfaces contracted. These surfaces are birational to the product of the …
We study the complexity of birational self-maps of a projective threefold $X$ by looking at the birational type of surfaces contracted. These surfaces are birational to the product of the projective line with a smooth projective curve. We prove that the genus of the curves occuring is unbounded if and only if $X$ is birational to a conic bundle or a fibration into cubic surfaces. Similarly, we prove that the gonality of the curves is unbounded if and only if $X$ is birational to a conic bundle.
We conjecture that the number of components of the fiber over infinity of Landau-Ginzburg model for a smooth Fano variety X equals the dimension of the anticanonical system of X.We …
We conjecture that the number of components of the fiber over infinity of Landau-Ginzburg model for a smooth Fano variety X equals the dimension of the anticanonical system of X.We verify this conjecture for log Calabi-Yau compactifications of toric Landau-Ginzburg models for smooth Fano threefolds, complete intersections in projective spaces, and some toric varieties.
We classify finite subgroups $G\subset\mathrm{PGL}_4(\mathbb{C})$ such that $\mathbb{P}^3$ is not $G$-birational to conic bundles and del Pezzo fibrations, and explicitly describe all $G$-Mori fibre spaces that are $G$-birational to $\mathbb{P}^3$ …
We classify finite subgroups $G\subset\mathrm{PGL}_4(\mathbb{C})$ such that $\mathbb{P}^3$ is not $G$-birational to conic bundles and del Pezzo fibrations, and explicitly describe all $G$-Mori fibre spaces that are $G$-birational to $\mathbb{P}^3$ for these subgroups.
We study equivariant birational geometry of (rational) quartic double solids ramified over (singular) Kummer surfaces.
We study equivariant birational geometry of (rational) quartic double solids ramified over (singular) Kummer surfaces.
We prove that every smooth divisor in $\mathbb{P}^1\times\mathbb{P}^1\times\mathbb{P}^2$ of degree $(1,1,2)$ is K-stable.
We prove that every smooth divisor in $\mathbb{P}^1\times\mathbb{P}^1\times\mathbb{P}^2$ of degree $(1,1,2)$ is K-stable.
We classify birational involutions of the real projective plane up to conjugation. In contrast with an analogous classification over the complex numbers (due to E. Bertini, G. Castelnuovo, F. Enriques, …
We classify birational involutions of the real projective plane up to conjugation. In contrast with an analogous classification over the complex numbers (due to E. Bertini, G. Castelnuovo, F. Enriques, L. Bayle and A. Beauville), which includes 4 different classes of involutions, we discover 12 different classes over the reals, and provide many examples when the fixed curve of an involution does not determine its conjugacy class in the real plane Cremona group.
We study degree of irrationality of quasismooth anticanonically embedded weighted Fano 3-fold hypersurfaces that have terminal singularities.
We study degree of irrationality of quasismooth anticanonically embedded weighted Fano 3-fold hypersurfaces that have terminal singularities.
We prove that all smooth Fano threefolds in the families 2.1, 2.2, 2.3, 2.4, 2.6 and 2.7 are K-stable, and we also prove that smooth Fano threefolds in the family …
We prove that all smooth Fano threefolds in the families 2.1, 2.2, 2.3, 2.4, 2.6 and 2.7 are K-stable, and we also prove that smooth Fano threefolds in the family 2.5 that satisfy one very explicit generality condition are K-stable.
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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Chapter O. Notation and preliminaries § 0-1.Kleiman's criterion for ampleness § 0-2.Definitions of terminal, canonical and (weak) log-terminal singularities § 0-3.Canonical varieties § 0-4.The minimal model conjecture Chapter 1. Vanishing …
Chapter O. Notation and preliminaries § 0-1.Kleiman's criterion for ampleness § 0-2.Definitions of terminal, canonical and (weak) log-terminal singularities § 0-3.Canonical varieties § 0-4.The minimal model conjecture Chapter 1. Vanishing theorems § 1-1.Covering Lemma § 1-2.Vanishing theorem of Kawamata and Viehweg § 1-3.Vanishing theorem of Elkik and Fujita Chapter 2. Non-Vanishing Theorem § 2-1.Non-Vanishing Theorem Chapter 3. Base Point Free Theorem § 3-1.Base Point Free Theorem § 3-2.Contractions of extremal faces § 3-3.Canonical rings of varieties of general type Chapter 4. Cone Theorem § 4-1.Rationality Theorem § 4-2.The proof of the Cone Theorem Chapter 5. Flip Conjecture § 5-1.Types of contractions of extremal rays § 5-2.Flips of toric morphisms Chapter 6. Abundance Conjecture § 6-1.Nef and abundant divisors
In the last chapter, we introduced the Noether–Fano method for proving nonrationality of certain varieties. In this chapter, we develop further practical techniques for carrying out this program.
In the last chapter, we introduced the Noether–Fano method for proving nonrationality of certain varieties. In this chapter, we develop further practical techniques for carrying out this program.
We compute global log canonical thresholds of some smooth Fano threefolds.Contents 1. Introduction 2. Preliminaries 3. The Mukai-Umemura threefold 4. Cubic surfaces 5. Del Pezzo surfaces 6. Toric varieties 7. …
We compute global log canonical thresholds of some smooth Fano threefolds.Contents 1. Introduction 2. Preliminaries 3. The Mukai-Umemura threefold 4. Cubic surfaces 5. Del Pezzo surfaces 6. Toric varieties 7. Del Pezzo threefolds 8. Fano threefolds with ρ = 2 9. Fano threefolds with ρ = 3 10.Fano threefolds with ρ 4 11.Upper bounds Appendix A. By Jean-Pierre Demailly.On Tian's invariant and log canonical thresholds Appendix B. The Big Table References U α ψ
Intermediate Jacobians of threefolds 1. Algebraic correspondences and homology relations.2. Families of algebraic curves on a threefold.3. The intermediate Jacobian and its polarizing class.4. The Abel-Jacobi mapping.Part Two.Geometry of cubic …
Intermediate Jacobians of threefolds 1. Algebraic correspondences and homology relations.2. Families of algebraic curves on a threefold.3. The intermediate Jacobian and its polarizing class.4. The Abel-Jacobi mapping.Part Two.Geometry of cubic hypersurfaces 5.The dual mapping, Lefschetz hypersurfaces.6. Cubic hypersurfaces.7. The variety of lines on a cubic hypersurface.Part Three.The cubic threefold 8.The Fano surface of lines on a cubic threefold, the double point case.9. A topological model for the Fano surface, the non-singular case.10. Distinguished divisors on the Fano surface.Part Four.The intermediate Jacobian of the cubic threefold 11.The Gherardelli-Todd isomorphism.12.The Gauss map and the tangent bundle theorem.13.The "double-six", Torelli, and irrationality theorems.Appendices A. Equivalence relations on the algebraic one-cycles lying on a cubic threefold.B. Unirationality.C. Mumford's theory of Prym varieties and a comment on moduli.
We introduce complex singularity exponents of plurisubharmonic functions and prove a general semi-continuity result for them. This concept contains as a special case several similar concepts which have been considered …
We introduce complex singularity exponents of plurisubharmonic functions and prove a general semi-continuity result for them. This concept contains as a special case several similar concepts which have been considered e.g. by Arnold and Varchenko, mostly for the study of hypersurface singularities. The plurisubharmonic version is somehow based on a reduction to the algebraic case, but it also takes into account more quantitative informations of great interest for complex analysis and complex differential geometry. We give as an application a new derivation of criteria for the existence of Kähler–Einstein metrics on certain Fano orbifolds, following Nadel's original ideas (but with a drastic simplication in the technique, once the semi-continuity result is taken for granted). In this way, three new examples of rigid Kähler–Einstein Del Pezzo surfaces with quotient singularities are obtained. Nous introduisons les exposants de singularités complexes des fonctions plurisousharmoniques et démontrons un théorème de semi-continuité général pour ceux-ci. Le concept étudié contient comme cas particulier des concepts voisins qui ont été considérés par exemple par Arnold et Varchenko, principalement pour l'étude des singularités d'hypersurfaces. La version plurisousharmonique repose en définitive sur une réduction au cas algébrique, mais elle prend aussi en compte des informations quantitatives d'un grand intérêt pour l'analyse complexe et la géométrie différentielle complexe. Nous décrivons en application une nouvelle approche des critères d'existence de métriques de Kähler–Einstein pour les variétés de Fano, en nous inspirant des idées originales de Nadel – mais avec des simplifications importantes de la technique, une fois que le résultat de semi-continuité est utilisé comme outil de base. Grâce à ces critères, nous obtenons trois nouveaux exemples de surfaces de Del Pezzo à singularités quotients, rigides, possédant une métrique de Kähler–Einstein.
The birational superrigidity and, in particular, the non-rationality of a smooth three-dimensional quartic was proved by V. Iskovskikh and Yu. Manin in 1971, and this led immediately to a counterexample …
The birational superrigidity and, in particular, the non-rationality of a smooth three-dimensional quartic was proved by V. Iskovskikh and Yu. Manin in 1971, and this led immediately to a counterexample to the three-dimensional Luroth problem. Since then, birational rigidity and superrigidity have been proved for a broad class of higher-dimensional varieties, among which the Fano varieties occupy the central place. The present paper is a survey of the theory of birationally rigid Fano varieties.
In this paper we prove that any birational mapping between smooth hypersurfaces of degree four is an isomorphism. Since B. Segre constructed examples of smooth unirational quartics, this leads to …
In this paper we prove that any birational mapping between smooth hypersurfaces of degree four is an isomorphism. Since B. Segre constructed examples of smooth unirational quartics, this leads to a negative resolution of the three-dimensional Lüroth problem. Bibliography: 13 items.
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We prove birational superrigidity of direct products $V=F_1\times...\times F_K$ of primitive Fano varieties of the following two types: either $F_i\subset{\mathbb P}^M$ is a general hypersurface of degree $M$, $M\geq 6$, …
We prove birational superrigidity of direct products $V=F_1\times...\times F_K$ of primitive Fano varieties of the following two types: either $F_i\subset{\mathbb P}^M$ is a general hypersurface of degree $M$, $M\geq 6$, or $F_i\stackrel{\sigma}{\to}{\mathbb P}^M$ is a general double space of index 1, $M\geq 3$. In particular, each structure of a rationally connected fiber space on $V$ is given by a projection onto a direct factor. The proof is based on the connectedness principle of Shokurov and Koll\' ar and the technique of hypertangent divisors.
A lower bound for global log canonical thresholds on smooth hypersurfaces is found. This bound cannot be improved for the fixed degree and dimension of the hypersurface.
A lower bound for global log canonical thresholds on smooth hypersurfaces is found. This bound cannot be improved for the fixed degree and dimension of the hypersurface.
In this paper, we study the birational geometry of certain examples of mildly singular quartic 3-folds. A quartic 3-fold is a special case of a Fano variety, that is, a …
In this paper, we study the birational geometry of certain examples of mildly singular quartic 3-folds. A quartic 3-fold is a special case of a Fano variety, that is, a variety X with ample anticanonical sheaf [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="01i" /]. Nonsingular Fano 3-folds have been studied extensively. Examples studied so far fall within two classes: either X is "close to being rational," and it then has very many biregularly distinct birational models as a Fano 3-fold, or, at the other extreme, X has a unique model. In this paper we construct examples of singular quartic 3-folds with exactly two birational models as Fano 3-folds; the other model is a complete intersection Y 3,4 ⊂ [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="02i" /](1, 1, 1, 1, 2, 2) of a quartic and a cubic in weighted projective space [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="03i" /](1 4 , 2 2 ).
1. Introduction Let X be an algebraic variety defined over a number field F . We will say that rational points are potentially dense if there exists a finite extension …
1. Introduction Let X be an algebraic variety defined over a number field F . We will say that rational points are potentially dense if there exists a finite extension K/F such that the set of K -rational points X ( K ) is Zariski dense in X . The main problem is to relate this property to geometric invariants of X . Hypothetically, on varieties of general type rational points are not potentially dense. In this paper we are interested in smooth projective varieties such that neither they nor their unramified coverings admit a dominant map onto varieties of general type. For these varieties it seems plausible to expect that rational points are potentially dense (see [2]).
We show that affine cones over smooth cubic surfaces do not admit non-trivial \mathbb{G}_{a} -actions.
We show that affine cones over smooth cubic surfaces do not admit non-trivial \mathbb{G}_{a} -actions.
A consistent exposition of the arguments and constructions of the method of maximal singularities, the aim of which is to describe birational iso/automorphisms of Fano varieties and Fano fibrations. The …
A consistent exposition of the arguments and constructions of the method of maximal singularities, the aim of which is to describe birational iso/automorphisms of Fano varieties and Fano fibrations. The principal elements of the method are considered: N{\" o}ther-Fano inequality, maximal cycles, infinitely near maximal singularities, exclusion and untwisting. In a detailed way the crucial technical points are discussed. We also give a new version of the proof of Sarkisov theorem which is ideologically more close to the original arguments of V.A.Iskovskikh and Yu.I.Manin.
Let $S \subset \P^n$ be a smooth quartic hypersurface defined over a number field $K$. If $n \ge 4$, then for some finite extension $K'$ of $K$ the set $S(K')$ …
Let $S \subset \P^n$ be a smooth quartic hypersurface defined over a number field $K$. If $n \ge 4$, then for some finite extension $K'$ of $K$ the set $S(K')$ of $K'$-rational points of $S$ is Zariski dense.
Letk be a perfect field of arbitrary characteristic. The main object of this paper is to establish some new objects associated with algebraic surfaces F defined overk which are invariants …
Letk be a perfect field of arbitrary characteristic. The main object of this paper is to establish some new objects associated with algebraic surfaces F defined overk which are invariants for birational transformations defined overk. There are two main applications. The first is that if K is any extension ofk of degree 2, then there are infinitely many birationally inequivalent rational surfaces defined overk which all become birationally equivalent to the plane over K. The second application is to a partial classification of the del Pezzo surfaces for birational equivalence overk. For our purposes a del Pezzo surface defined overk is a nonsingular rational surface with a very ample anticanonical system, so the nonsingular cubic surfaces are a special care. As we use the language of schemes, we have to prove some classical results in the new framework, notably some results of Enriques [7] on the classification of rational surfaces. In the last section we produce evidence for the conjecture that if the fieldk is quasialgebraically closed (in the sense of Lang [11]), then a rational surface defined overk always has a point on it defined overk.
In 1934, DuVal [3] listed the configurations of curves which can be obtained by resolving certain isolated double points of embedded surfaces (they are depicted in the figure below). These …
In 1934, DuVal [3] listed the configurations of curves which can be obtained by resolving certain isolated double points of embedded surfaces (they are depicted in the figure below). These configurations arise naturally in other contexts, for instance as exceptional curves for pluricanonical embeddings of surfaces [7], and so it seems desirable to have a converse result, showing that a singularity giving rise to such a configuration is necessarily a double point. We have reconsidered the question in a more general context, and obtain in addition the correct nunferical characterization of singularities (cf. definition below). This characterization is made without the assumption that the surface is embedded and points out the connection of Du Val's work with Castelnuovo's criterion for exceptional curves [2], ([8], p. 38). Finally, we list the configurations obtained from rational triple points.
is bijective for i < n - 1 and surjective for i = n - 1. Several proofs of this theorem are to be found in the literature (see [5] …
is bijective for i < n - 1 and surjective for i = n - 1. Several proofs of this theorem are to be found in the literature (see [5] for an account of the problem). Recently Thom has given a proof (unpublished) which, as far as we know, is the first to use Morse's theory of critical points. We present in ? 3, in a slightly more general setting, an alternate proof inspired by Thom's discovery. Our statement is given in the equivalent language of cohomology. The proof is derived from a theorem on Stein manifolds which is presented in ? 2. Some standard properties of the distance function which we require are assembled in ? 1 for the sake of completeness.
It is known that the -ޑfactoriality of a nodal quartic 3-fold in ސ 4 implies its nonrationality.We prove that a nodal quartic 3-fold with at most 8 nodes is -ޑfactorial, …
It is known that the -ޑfactoriality of a nodal quartic 3-fold in ސ 4 implies its nonrationality.We prove that a nodal quartic 3-fold with at most 8 nodes is -ޑfactorial, while one with 9 nodes is not -ޑfactorial if and only if it contains a plane.There are nonrational non--ޑfactorial nodal quartic 3-folds.In particular, we prove the nonrationality of a general non--ޑfactorial nodal quartic 3-fold that contains either a plane or a smooth del Pezzo surface of degree 4.
Let S be a surface in complex projective 3-space, having only nodes as singularities. Suppose that S has degree 6. We show that the maximum number of nodes which S …
Let S be a surface in complex projective 3-space, having only nodes as singularities. Suppose that S has degree 6. We show that the maximum number of nodes which S can have is 65. An abbreviated history of this is as follows. Basset showed that S can have at most 66 nodes. Catanese and Ceresa and Stagnaro constructed sextic surfaces having 64 nodes. Barth has recently exhibited a 65 node sextic surface. We complete the story by showing that S cannot have 66 nodes. Let f: S~ --> S be a minimal resolution of singularities. A set N of nodes on S is even if there exists a divisor Q on S~ such that 2Q ~ f^{-1}(N). We show that a nonempty even set of nodes on S must have size 24, 32, 40, 56, or 64. This result is key to showing the nonexistence of the 66 node sextic. We do not know if a sextic surface can have an even node set of size 56 or 64. The existence or nonexistence of large even node sets is related to the following vanishing problem. Let S be a normal surface of degree s in CP^3. Let D be a Weil divisor on S such that D is Q-rationally equivalent to rH, for some r \in \Q. Under what circumstances do we have H^1(O_S(D)) = 0? For instance, this holds when r < 0. For s=4 and r=0, H^1 can be nonzero. For s=6 and r=0, if a 56 or 64 node even set exists, then H^1 can be nonzero. The vanishing of H^1 is also related to linear normality, quadric normality, etc. of set-theoretic complete intersections in P^3.
In this paper the author determines when the principally polarized Prymian of a Beauville pair satisfying a certain stability type condition is isomorphic to the Jacobian of a nonsingular curve. …
In this paper the author determines when the principally polarized Prymian of a Beauville pair satisfying a certain stability type condition is isomorphic to the Jacobian of a nonsingular curve. As an application, he points out new components in the Andreotti-Mayer variety of principally polarized abelian varieties of dimension whose theta-divisors have singular locus of dimension ; he also proves a rationality criterion for conic bundles over a minimal rational surface in terms of the intermediate Jacobian. The first part of the paper contains the necessary preliminary material introducing the reader to the modern theory of Prym varieties.Figures: 10. Bibliography: 32 titles.
We prove the $\mathbb {Q}$-factoriality of a nodal hypersurface in $\mathbb {P}^{4}$ of degree $n$ with at most ${\frac {(n-1)^{2}}{4}}$ nodes and the $\mathbb {Q}$-factoriality of a double cover of …
We prove the $\mathbb {Q}$-factoriality of a nodal hypersurface in $\mathbb {P}^{4}$ of degree $n$ with at most ${\frac {(n-1)^{2}}{4}}$ nodes and the $\mathbb {Q}$-factoriality of a double cover of $\mathbb {P}^{3}$ branched over a nodal surface of degree $2r$ with at most ${\frac {(2r-1)r}{3}}$ nodes.