We propose a refined but natural notion of toric degenerations that respect a given embedding and show that within this framework a Gorenstein Fano variety can only be degenerated to a Gorenstein Fano toric variety if it is embedded via its anticanonical embedding. This also gives a precise criterion for reflexive polytopes to appear, which might be required for applications in mirror symmetry. For the proof of this statement we will study polytopes whose polar dual is a lattice polytope. As a byproduct we generalize a connection between the number of lattice points in a rational convex polytope and the Euler characteristic of an associated torus invariant rational Weil divisor, allowing us to show that Ehrhart-Macdonald Reciprocity and Serre Duality are equivalent statements for a broad class of varieties. Additionally, we conjecture a necessary and sufficient condition for the Ehrhart quasi-polynomial of a rational convex polytope to be a polynomial. Finally, we show that the anticanonical line bundle on a Gorenstein Fano variety with at worst rational singularities is uniquely determined by a combinatorial condition of its Hilbert polynomial.
Given a reductive group $G$ and a parabolic subgroup $P\subset G$, with maximaltorus $T$, we consider (following Dabrowski's work) the closure $X$ of a generic $T$-orbit in $G/P$, and determine in combinatorial termswhen the toric variety $X$ is $\mathbb{Q}$-Gorenstein Fano, extending in this way the classification of smooth Fano generic closures given by Voskresenski\uı and Klyachko. As an application, we apply the well known correspondence between Gorenstein Fano toric varieties and reflexive polytopes in order to exhibit which reflexive polytopes correspond to generic closures -- this list …
We give a characterization of Gorenstein toric Fano varieties of dimension $n$ with index $n$ among toric varieties. As an application, we give a strong version of Fujita's freeness conjecture and also give a simple proof of Fujita's very ampleness conjecture on Gorenstein toric varieties.
We investigate Gorenstein toric Fano varieties by combinatorial methods using the notion of a reflexive polytope which appeared in connection to mirror symmetry. The paper contains generalisations of tools and previously known results for nonsingular toric Fano varieties. As applications we obtain new classification results, bounds of invariants and formulate conjectures concerning combinatorial and geometrical properties of reflexive polytopes.
We consider a surface that admits a $\mathbb{Q}$-Gorenstein degeneration to a cyclic quotient singularity $\frac{1}{dn^2}(1,dna-1)$. Under several technical assumptions, we construct $d$ exceptional vector bundles of rank $n$ which are orthogonal to each other.
We give a characterizaion of Gorenstein toric Fano $n$-folds with index $n-1$, which is called Gorenstein toric Del Pezzo $n$-folds, among toric varieties. In practice, we obtain a condition for a lattice $n$-polytope to be a Gorenstein Fano polytope. In our proof, we do not use Batyrev-Juny's classification of Gorenstein Fano polytopes.
We obtain a combinatorial description of Gorenstein spherical Fano varieties in terms of certain polytopes, generalizing the combinatorial description of Gorenstein toric Fano varieties by reflexive polytopes and its extension to Gorenstein horospherical Fano varieties due to Pasquier. Using this description, we show that the rank of the Picard group of an arbitrary $d$-dimensional $\mathbb{Q}$-factorial Gorenstein spherical Fano variety is bounded by $2d$. This paper also contains an overview of the description of the natural representative of the anticanonical divisor class of a spherical va…
We classify the non-toric, $\mathbb Q$-factorial, Gorenstein, log terminal Fano threefolds of Picard number one that admit an effective action of a two-dimensional algebraic torus.
Q-factorial Gorenstein toric Fano varieties X of dimension d with Picard number rho(X) correspond to simplicial reflexive d-polytopes with rho(X)+d vertices. Casagrande showed that any simplicial reflexive d-polytope has at most 3d vertices, if d is even, respectively, 3d-1, if d is odd. Moreover, it is known that equality for d even implies uniqueness up to unimodular equivalence. In this paper we completely classify all simplicial reflexive d-polytopes having 3d-1 vertices, corresponding to d-dimensional Q-factorial Gorenstein toric Fano varieties with Picard number 2d-1. For d even, there e…
We classify some special classes of non-rational Fano threefolds with terminal singularities. In particular, all such hyperelliptic and trigonal varieties are found.
We obtain a classification of a Q-factorial Q-Fano 3-fold $X$ with the following properties: the Picard number of $X$ is 1; the Gorenstein index of $X$ is 2; the Fano index of $X$ is 1/2; $h^0 (-K_X) \geq 4$; there exists an index 2 point $P$ such that $(X,P)\simeq (\{xy +f(z^2,u)=0\} / \Bbb Z_2 (1,1,1,0), o)$ with $ord f(Z,U)=1$.
We classify nonrational Fano threefolds $X$ with terminal Gorenstein singularities such that $\mathrm{\rk}\, \mathrm{\Pic}(X)=1$, $(-K_X)^3\ge 8$, and $\mathrm{\rk}\, \mathrm{\Cl}(X)\le 2$.
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.
In this paper we mainly describe $\mathbb{Q}$-Gorenstein smoothings of projective surfaces with only Wahl singularities which have birational fibers. For instance, these degenerations appear in normal degenerations of the projective plane, and in boundary divisors of the KSBA compactification of the moduli space of surfaces of general type [KSB88]. We give an explicit description of them as smooth deformations plus 3-fold birational operations, through the flips and divisorial contractions in [HTU13]. We interpret the continuous part (smooth deformations) as degenerations of certain curves in …
We show that finitely generated Cox rings are Gorenstein. This leads to a refined characterization of varieties of Fano type: they are exactly those projective varieties with Gorenstein canonical quasicone Cox ring. We then show that for varieties of Fano type and Kawamata log terminal quasicones, iteration of Cox rings is finite with factorial master Cox ring. Moreover, we prove a relative version of Cox ring iteration for almost principal solvable $G$-bundles.
There exist exactly 166 4-dimensional reflexive polytopes such that the corresponding 4-dimensional Gorenstein toric Fano varieties have at worst terminal singularities in codimension 3 and their anticanonical divisor is divisible by 2. For every such a polytope, one naturally obtains a family of Fano hypersurfaces X with at worst conifold singularities. A generic 3-dimensional Fano hypersurface X can be interpreted as a flat conifold degeneration of some smooth Fano 3-folds Y whose classification up to deformation was obtained by Iskovskikh, Mori and Mukai. In this case, both Fano varieties X…
The correspondence between Gorenstein Fano toric varieties and reflexive polytopes has been generalized by Ilten and Süß to a correspondence between Gorenstein Fano complexity-one $T$-varieties and Fano divisorial polytopes. Motivated by the finiteness of reflexive polytopes in fixed dimension, we show that over a fixed base polytope, there are only finitely many Fano divisorial polytopes, up to equivalence. We classify two-dimensional Fano divisorial polytopes, recovering Huggenberger's classification of Gorenstein del Pezzo $\mathbb{K}^*$-surfaces. Furthermore, we show that any three-dimensi…
We construct some new deformation families of four-dimensional Fano manifolds of index $1$ in some known classes of Gorenstein formats. These families have explicit descriptions in terms of equations, defining their image under the anti-canonical embedding in some weighted projective space. The constructed families have relatively smaller anti-canonical degrees than most other known families of smooth Fano 4-folds.
James Ax conjectured that every pseudo algebraically closed field is $C_1$. We prove this conjecture in characteristic 0 by relating it to degenerations of Fano varieties.
We consider Fano threefolds $X$ with canonical Gorenstein singularities. Under additional assumption that $X$ has at least one non-cDV point we prove a sharp bound of the degree: $-K_X^3\le 72$.
We propose a combinatorical duality for lattice polyhedra which conjecturally gives rise to the pairs of mirror symmetric families of Calabi-Yau complete intersections in toric Fano varieties with Gorenstein singularities. Our construction is a generalization of the polar duality proposed by Batyrev for the case of hypersurfaces.
Let $X$ be a Gorenstein canonical Fano variety of coindex $4$ and dimension $n$ with $H$ fundamental divisor. Assume $h^0(X, H) \geq n -2$. We prove that a general element of the linear system $|mH|$ has at worst canonical singularities for any integer $m \geq 1$. When $X$ has terminal singularities and $n \geq 5$, we show that a general element of $|mH|$ has at worst terminal singularities for any integer $m \geq 1$. When $n=4$, we give an example of Gorenstein terminal Fano fourfold $X$ such that a general element of $|H|$ does not have terminal singularities.
In this paper, we give a tropical method for computing Gromov-Witten type invariants of Fano manifolds of special type. This method applies to those Fano manifolds which admit toric degenerations to toric Fano varieties with singularities allowing small resolutions. Examples include (generalized) flag manifolds of type A, and some moduli space of rank two bundles on a genus two curve.
We show that every rank one smooth Fano threefold has a weak Landau-Ginzburg model coming from a toric degeneration. The fibers of these Landau-Ginzburg models can be compactified to K3 surfaces with Picard lattice of rank 19. We also show that any smooth Fano variety of arbitrary dimension which is a complete intersection of Cartier divisors in weighted projective space has a very weak Landau-Ginzburg model coming from a toric degeneration.
For fixed degree $d\leq 12$, we study the Hilbert scheme of degree $d$ smooth Fano threefolds in their anticanonical embeddings. We use this to classify all possible degenerations of these varieties to toric Fano varieties with at most canonical Gorenstein singularities.
We prove that K-polystable degenerations of Q-Fano varieties are unique. Furthermore, we show that the moduli stack of K-stable Q-Fano varieties is separated. Together with [Jia17,BL18], the latter result yields a separated Deligne-Mumford stack parametrizing all uniformly K-stable Q-Fano varieties of fixed dimension and volume. The result also implies that the automorphism group of a K-stable Q-Fano variety is finite.
We construct degenerations of Mukai varieties and linear sections thereof to special unobstructed Fano Stanley-Reisner schemes corresponding to convex deltahedra. This can be used to find toric degenerations of rank one index one Fano threefolds. In the second part we find many higher dimensional unobstructed Fano and Calabi-Yau Stanley-Reisner schemes. The main result is that the Stanley-Reisner ring of the boundary complex of the dual polytope of the associahedron has trivial $T^2$.
This paper studies the defect of terminal Gorenstein Fano 3 folds. I determine a bound on the defect of terminal Gorenstein Fano 3-folds of Picard rank 1 that do not contain a plane. I give a general bound for quartic 3-folds and indicate how to study the defect of terminal Gorenstein Fano 3-folds with Picard rank 1 that contain a plane.
Consider a family of Fano varieties $π: X \longrightarrow B\ni o$ over a curve germ with a smooth total space $X$. Assume that the generic fiber is smooth and the special fiber $F=π^{-1}(o)$ has simple normal crossings. Then $F$ is called a semistable degeneration of Fano varieties. We show that the dual complex of $F$ is a simplex of dimension $\leq \mathrm{dim}\ F$. Simplices of any admissible dimension can be realized for any dimension of the fiber. Using this result and the Minimal Model Program in dimension $3$ we reproduce the classification of the semistable degenerations of del Pezzo s…
In this Thesis, I investigate how Fano manifolds equipped with a Kahler-Einstein metric can degenerate as metric spaces (in the Gromov-Hausdorff topology) and some of the relations of this question with Algebraic Geometry, in particular in the direction of the study of moduli spaces and their compactifications.
We prove that for any $\mathbb{Q}$-Fano variety $X$, the special $\mathbb{R}$-test configuration that minimizes the $H$-functional is unique and has a K-semistable $\mathbb{Q}$-Fano central fibre $(W, ξ)$. Moreover there is a unique K-polystable degeneration of $(W, ξ)$. As an application, we confirm the conjecture of Chen-Sun-Wang about the algebraic-uniqueness for Kähler-Ricci flow limits on Fano manifolds, which implies that the Gromov-Hausdorff limit of the flow does not depend on the choice of initial Kähler metrics. The results are achieved by studying algebraic optimal degeneration prob…
For an arbitrary smooth n-dimensional Fano variety $X$ we introduce the notion of a small toric degeneration. Using small toric degenerations of Fano n-folds $X$, we propose a general method for constructing mirrors of Calabi-Yau complete intersections in $X$. Our mirror construction is based on a generalized monomial-divisor mirror correspondence which can be used for computing Gromov-Witten invariants of rational curves via specializations of GKZ-hypergeometric series.
We consider the space KR(n,F) of Kahler-Ricci solitons on n-dimensional Fano manifolds with Futaki invariant bounded by F. We prove a partial C^0 estimate for KR(n,F) as a generalization of the recent work of Donaldson-Sun for Fano Kahler-Einstein manifolds. In particular, any sequence in KR(n,F) has a convergent subsequence in the Gromov-Hausdorff topology to a Kahler- Ricci soliton on a Q-Fano variety with log terminal singularities.
We classified prime $\mathbb{Q}$-Fano $3$-folds $X$ with only $1/2(1,1,1)$-singularities and with $h^{0}(-K_{X})\geq 4$ a long time ago. The classification was undertaken by blowing up each $X$ at one $1/2(1,1,1)$-singularity and constructing a Sarkisov link. The purpose of this paper is to reveal the geometries behind the Sarkisov links for $X$ in 5 classes. The main result asserts that any $X$ in the 5 classes can be embedded as linear sections into bigger dimensional $\mathbb{Q}$-Fano varieties called key varieties, where the key varieties are constructed by extending partially the Sarkisov…
We establish an algebraic approach to prove the properness of moduli spaces of K-polystable Fano varieties and reduce the problem to a conjecture on destabilizations of K-unstable Fano varieties. Specifically, we prove that if the stability threshold of every K-unstable Fano variety is computed by a divisorial valuation, then such K-moduli spaces are proper. The argument relies on studying certain optimal destabilizing test configurations and constructing a Theta-stratification on the moduli stack of Fano varieties.
In the paper "Birational geometry via moduli spaces" by I. Cheltsov, L. Katzarkov, and V. Przyjalkowski a new structure connecting toric degenerations of smooth Fano threefolds by projections was introduced; using Mirror Symmetry these connections were transferred to the side of Landau--Ginzburg models. In the paper mentioned above a nice way to connect of Picard rank one Fano threefolds was found. We apply this approach to all smooth Fano threefolds, connecting their degenerations by toric basic links. In particular, we find a lot of Gorenstein toric degenerations of smooth Fano threefolds we…
For an appropriate class of Fano complete intersections in toric varieties, we prove that there is a concrete relationship between degenerations to specific toric subvarieties and expressions for Givental's Landau-Ginzburg models as Laurent polynomials. As a result, we show that Fano varieties presented as complete intersections in partial flag manifolds admit degenerations to Gorenstein toric weak Fano varieties, and their Givental Landau-Ginzburg models can be expressed as corresponding Laurent polynomials. We also use this to show that all of the Laurent polynomials obtained by Coates, Kasp…
Gorenstein Fano polytopes arising from finite partially ordered sets will be introduced. Then we study the problem which partially ordered sets yield smooth Fano polytopes.
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 singularities which are not intersections of quadrics. We also study the rationality questions for most of these varieties.
