Fano Varieties Research Articles

Moduli of surfaces in

The main goal of this paper is to construct a compactification of the moduli space of degree $d \geqslant 5$ surfaces in $\mathbb {P}^{3}_{{{\mathbb {C}}}}$, i.e. a parameter space whose interior points correspond to (equivalence classes of) smooth surfaces in $\mathbb {P}^{3}$ and whose boundary points correspond to degenerations of such surfaces. We consider a divisor $D$ on a Fano variety $Z$ as a pair $(Z, D)$ satisfying certain properties. We find a modular compactification of such pairs and, in the case of $Z = {{\mathbb {P}}}^{3}$ and $D$ a surface, use their properties to classify the pairs on the boundary of the moduli space.

Motivic Limits for Fano Varieties of k-Planes

Abstract We study the probability that an $(n - m)$-dimensional linear subspace in $\mathbb{P}^n$ or a collection of points spanning such a linear subspace is contained in an m-dimensional variety $Y \subset \mathbb{P}^n$. This involves a strategy used by Galkin–Shinder to connect properties of a cubic hypersurface to its Fano variety of lines via cut-and-paste relations in the Grothendieck ring of varieties. Generalizing this idea to varieties of higher codimension and degree, we can measure growth rates of weighted probabilities of k-planes contained in a sequence of varieties with varying initial parameters over a finite field. In the course of doing this, we move an identity motivated by rationality problems involving cubic hypersurfaces to a motivic statistics setting associated with cohomological stability.

Restricted Tangent Bundles for General Free Rational Curves

Abstract Suppose that $X$ is a smooth projective variety and that $C$ is a general member of a family of free rational curves on $X$. We prove several statements showing that the Harder–Narasimhan filtration of $T_{X}|_{C}$ is approximately the same as the restriction of the Harder–Narasimhan filtration of $T_{X}$ with respect to the class of $C$. When $X$ is a Fano variety, we prove that the set of all restricted tangent bundles for general free rational curves is controlled by a finite set of data. We then apply our work to analyze Peyre’s “freeness” formulation of Manin’s Conjecture in the setting of rational curves.

Fano manifolds with Lefschetz defect 3

Let X be a smooth, complex Fano variety, and δX its Lefschetz defect. By [4], if δX≥4, then X≅S×T, where dim⁡T=dim⁡X−2. In this paper we prove a structure theorem for the case where δX=3. We show that there exists a smooth Fano variety T with dim⁡T=dim⁡X−2 such that X is obtained from T with two possible explicit constructions; in both cases there is a P2-bundle Z over T such that X is the blow-up of Z along three pairwise disjoint smooth, irreducible, codimension 2 subvarieties. Then we apply the structure theorem to Fano 4-folds, to the case where X has Picard number 5, and to Fano varieties having an elementary divisorial contraction sending a divisor to a curve. In particular we complete the classification of Fano 4-folds with δX=3, started in [6].

On the birational motive of hyper-Kähler varieties

We introduce a new ascending filtration, that we call the co-radical filtration in analogy with the basic theory of co-algebras, on the Chow groups of pointed smooth projective varieties. In the case of zero-cycles on projective hyper-Kähler manifolds, we conjecture it agrees with a filtration introduced by Voisin. This is established for moduli spaces of stable objects on K3 surfaces, for generalized Kummer varieties and for the Fano variety of lines on a smooth cubic fourfold. Our overall strategy is to view the birational motive of a smooth projective variety as a co-algebra object with respect to the diagonal embedding and to show in the aforementioned cases the existence of a so-called strict grading whose associated filtration agrees with the filtration of Voisin. As results of independent interest, we upgrade to rational equivalence Voisin's notion of “surface decomposition” and show that the birational motive of some projective hyper-Kähler manifolds is determined, as a co-algebra object, by the birational motive of a surface. We also relate our co-radical filtration on the Chow groups of abelian varieties to Beauville's eigenspace decomposition.

Singularities of Fano varieties of lines on singular cubic fourfolds

Let $X$ be a cubic fourfold that has only simple singularities and does not contain a plane. We prove that the Fano variety of lines on $X$ has the same analytic type of singularity as the Hilbert scheme of two points on a surface with only ADE-singularities.

Positivity of the CM line bundle for K-stable log Fanos

We prove the bigness of the Chow–Mumford line bundle associated to a Q \mathbb {Q} -Gorenstein family of log Fano varieties of maximal variation with uniformly K-stable general geometric fibers. This result generalizes a theorem of Codogni and Patakfalvi to the logarithmic setting.

Laurent polynomials in Mirror Symmetry: why and how?

We survey the approach to mirror symmetry via Laurent polynomials, outlining some of the main conjectures, problems, and questions related to the subject. We discuss: how to construct Landau–Ginzburg models for Fano varieties; how to apply them to classification problems; and how to compute invariants of Fano varieties via Landau–Ginzburg models.

On Uniform K-Stability for Some Asymptotically log del Pezzo Surfaces

Motivated by the problem of the existence of Kähler–Einstein edge metrics, Cheltsov and Rubinstein conjectured the K-polystability of asymptotically log Fano varieties with small cone angles when the anti-log-canonical divisors are not big. Cheltsov, Rubinstein and Zhang proved it affirmatively in dimension 2 with irreducible boundaries except for the type (I.9B.$n$) with $1 \leq n \leq 6$. Unfortunately, Fujita, Liu, Süß, Zhang and Zhuang recently showed the non-K-polystability for some members of type (I.9B.1) and for some members of type (I.9B.2). In this article we show that Cheltsov–Rubinstein’s problem is true for all of the remaining cases. More precisely, we explicitly compute the deltainvariant for asymptotically log del Pezzo surfaces of type (I.9B.$n$) for all $n \geq 1$ with small cone angles. As a consequence, we finish Cheltsov–Rubinstein’s problem in dimension 2 with irreducible boundaries.

Generalized Kähler-Ricci flow on toric Fano varieties

We study the generalized Kähler-Ricci flow with initial data of symplectic type, and show that this condition is preserved. In the case of a Fano background with toric symmetry, we establish global existence of the normalized flow. We derive an extension of Perelman’s entropy functional to this setting, which yields convergence of nonsingular solutions at infinity. Furthermore, we derive an extension of Mabuchi’s K K -energy to this setting, which yields weak convergence of the flow.

Virasoro conjecture for the stable pairs descendent theory of simply connected 3-folds (with applications to the Hilbert scheme of points of a surface).

This paper concerns the recent Virasoro conjecture for the theory of stable pairs on a 3‐fold proposed by Oblomkov, Okounkov, Pandharipande, and the author. Here we extend the conjecture to 3‐folds with non‐(p,p)‐cohomology and we prove it in two specializations. For the first specialization, we let S be a surface with H1(S)=0 and consider the moduli space Pn(S×P1,n[P1]), which happens to be isomorphic to the Hilbert scheme S[n] of n points on S. The Virasoro constraints for stable pairs, in this case, can be formulated entirely in terms of descendents in the Hilbert scheme of points. The two main ingredients of the proof are the toric case and the existence of universal formulas for integrals of descendents on S[n]. The second specialization consists in taking the 3‐fold X to be a cubic and the curve class β to be the line class. In this case we compute the full theory of stable pairs using the geometry of the Fano variety of lines.

Hilbert series, machine learning, and applications to physics

We describe how simple machine learning methods successfully predict geometric properties from Hilbert series (HS). Regressors predict embedding weights in projective space to ∼1 mean absolute error, whilst classifiers predict dimension and Gorenstein index to >90% accuracy with ∼0.5% standard error. Binary random forest classifiers managed to distinguish whether the underlying HS describes a complete intersection with high accuracies exceeding 95%. Neural networks (NNs) exhibited success identifying HS from a Gorenstein ring to the same order of accuracy, whilst generation of “fake” HS proved trivial for NNs to distinguish from those associated to the three-dimensional Fano varieties considered.

On the automorphisms of Mukai varieties

Mukai varieties are Fano varieties of Picard number one and coindex three. In genus seven to ten they are linear sections of some special homogeneous varieties. We describe the generic automorphism groups of these varieties. When they are expected to be trivial for dimensional reasons, we show they are indeed trivial, up to three interesting and unexpected exceptions in genera 7, 8, 9, and codimension 4, 3, 2 respectively. We conclude in particular that a generic prime Fano threefold of genus g has no automorphisms for \(7\leqslant g\leqslant 10\). In the Appendix by Y. Prokhorov, the latter statement is extended to \(g=12\).

Rigid Gorenstein toric Fano varieties arising from directed graphs

A directed edge polytope {mathcal {A}}_G is a lattice polytope arising from root system A_n and a finite directed graph G. If every directed edge of G belongs to a directed cycle in G, then {mathcal {A}}_G is terminal and reflexive, that is, one can associate this polytope to a Gorenstein toric Fano variety X_G with terminal singularities. It is shown by Totaro that a toric Fano variety which is smooth in codimension 2 and {mathbb {Q}}-factorial in codimension 3 is rigid. In the present paper, we classify all directed graphs G such that X_G is a toric Fano variety which is smooth in codimension 2 and {mathbb {Q}}-factorial in codimension 3.

Derived categories of centrally‐symmetric smooth toric Fano varieties

AbstractWe exhibit full exceptional collections of vector bundles on any smooth, Fano arithmetic toric variety whose split fan is centrally symmetric

On the body of ample angles of asymptotically log Fano varieties

In dimension two, we reduce the classification problem for asymptotically log Fano pairs to the problem of determining generality conditions on certain blow-ups. In any dimension, we prove the rationality of the body of ample angles of an asymptotically log Fano pair, i.e., these convex bodies are always rational polytopes.

Valuative stability of polarised varieties

Fujita and Li have given a characterisation of K-stability of a Fano variety in terms of quantities associated to valuations, which has been essential to all recent progress in the area. We introduce a notion of valuative stability for arbitrary polarised varieties, and show that it is equivalent to K-stability with respect to test configurations with integral central fibre. The numerical invariant governing valuative stability is modelled on Fujita’s beta -invariant, but includes a term involving the derivative of the volume. We give several examples of valuatively stable and unstable varieties, including the toric case. We also discuss the role that the delta -invariant plays in the study of valuative stability and K-stability of polarised varieties.

Openness of uniform K-stability in families of $Q$-Fano varieties

Openness of uniform K-stability in families of $Q$-Fano varieties

Fano and Weak Fano Hessenberg Varieties

Regular semisimple Hessenberg varieties are smooth subvarieties of the flag variety, and their examples contain the flag variety itself and the permutohedral variety, which is a toric variety. We give a complete classification of Fano and weak Fano regular semisimple Hessenberg varieties in type A in terms of combinatorics of Hessenberg functions. In particular, we show that if the anticanonical bundle of a regular semisimple Hessenberg variety is nef, then it is in fact nef and big.

Equivariant completions of affine spaces

We survey recent results on open embeddings of the affine space $\mathbb{C}^n$ into a complete algebraic variety $X$ such that the action of the vector group $\mathbb{G}_a^n$ on $\mathbb{C}^n$ by translations extends to an action of $\mathbb{G}_a^n$ on $X$. We begin with the Hassett-Tschinkel correspondence describing equivariant embeddings of $\mathbb{C}^n$ into projective spaces and present its generalization for embeddings into projective hypersurfaces. Further sections deal with embeddings into flag varieties and their degenerations, complete toric varieties, and Fano varieties of certain types. Bibliography: 109 titles.