Fano Index Research Articles

On subvarieties of degenerations of Fano varieties

The goal of this work is to study geometric properties of geometrically irreducible subschemes on degenerations of Fano varieties (more generally, of separably rationally connected varieties). It is known that these geometrically irreducible subschemes exist when the ground field has characteristic zero or contains an algebraically closed subfield. We show that the dimension of this geometrically irreducible subscheme has a lower bound by the Fano index of the generic fiber.

Fano varieties with conjecturally largest Fano index

For Fano varieties of various singularities such as canonical and terminal, we construct examples with large Fano index growing doubly exponentially with dimension. By low-dimensional evidence, we conjecture that our examples have the largest Fano index for all dimensions.

On the birational geometry of $${\mathbb {Q}}$$-Fano threefolds of large Fano index, I

On the birational geometry of $${\mathbb {Q}}$$-Fano threefolds of large Fano index, I

The graded ring database for Fano 3-folds and the Bogomolov stability bound

My paper (Suzuki 2003) produced some computer routines in Magma (Bosma et al. J Symb Comp 24:235–265, 1997) for the numerical invariants of Fano 3-folds, and used them in particular to determine the maximum value f=19\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$f=19$$\\end{document} of the Fano index. As a byproduct of the research, extensive data associated with all possible sets of singular points of Fano 3-folds with Fano indices greater than or equal to 2 was obtained. Collaborative research with Gavin Brown developed an improved version of the Magma program. The data discussed above was added to the Graded Ring Data Base (Brown et al. 2015) at the University of Kent. Subsequently, GRDB, now located to the University of Warwick, recently modified its interface to accommodate additional conditions, facilitating a more refined selection of Fano manifolds. In this context, we focus on the inequality known as the Bogomolov stability bound. We present a list of candidates for Fano 3-folds that do not satisfy these conditions and propose the conjecture that they do not exist.This result has been independently obtained in Liu and Liu (2023).

Boundedness of finite morphisms onto Fano manifolds with large Fano index

Let f:Y→X be a finite morphism between Fano manifolds Y and X such that the Fano index of X is greater than 1. On the one hand, when both X and Y are fourfolds of Picard number 1, we show that the degree of f is bounded in terms of X and Y unless X≅P4; hence, such X does not admit any non-isomorphic surjective endomorphism. On the other hand, when X=Y is either a fourfold or a del Pezzo manifold, we prove that, if f is an int-amplified endomorphism, then X is toric. Moreover, we classify all the singular quadrics admitting non-isomorphic endomorphisms.

Weak Approximation for Fano Complete Intersections in Positive Characteristic

For a smooth curve B over a field k=k ¯ with char(k)=p, for every complete intersection X B in B× Speck ℙ k n of type (d 1 ,⋯,d c ), we prove weak approximation of adelic points of X B by k(B)-points at all places of (strong) potentially good reduction, if the Fano index is ≥2 and if p>max(d 1 ,⋯,d c ). This also applies to specializations of complex Fano manifolds with Picard rank 1 and Fano index 1 away from “bad primes”.

On K‐stability of some del Pezzo surfaces of Fano index 2

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.

K-stability of Fano varieties via admissible flags

Abstract We develop a general approach to prove K-stability of Fano varieties. The new theory is used to (a) prove the existence of Kähler-Einstein metrics on all smooth Fano hypersurfaces of Fano index two, (b) compute the stability thresholds for hypersurfaces at generalised Eckardt points and for cubic surfaces at all points, and (c) provide a new algebraic proof of Tian’s criterion for K-stability, amongst other applications.

FANO HYPERSURFACES WITH ARBITRARILY LARGE DEGREES OF IRRATIONALITY

We show that complex Fano hypersurfaces can have arbitrarily large degrees of irrationality. More precisely, if we fix a Fano index $e$, then the degree of irrationality of a very general complex Fano hypersurface of index $e$and dimension n is bounded from below by a constant times $\sqrt{n}$. To our knowledge, this gives the first examples of rationally connected varieties with degrees of irrationality greater than 3. The proof follows a degeneration to characteristic$p$argument, which Kollár used to prove nonrationality of Fano hypersurfaces. Along the way, we show that in a family of varieties, the invariant ‘the minimal degree of a dominant rational map to a ruled variety’ can only drop on special fibers. As a consequence, we show that for certain low-dimensional families of varieties, the degree of irrationality also behaves well under specialization.

ℚ-Fano threefolds of index 7

We show that, for a $\mathbb Q$-Fano threefold $X$ of Fano index 7, the inequality $\dim |-K_X| \ge 15$ implies that $X$ is isomorphic to one of the following varieties $\mathbb P (1^2,2,3)$, $X_6 \subset \mathbb P (1,2^2,3,5)$ or $X_6 \subset \mathbb P (1,2,3^2,4)$.

On the Fukaya category of a Fano hypersurface in projective space

This paper is about the Fukaya category of a Fano hypersurface $X \subset \mathbb{CP}^n$. Because these symplectic manifolds are monotone, both the analysis and the algebra involved in the definition of the Fukaya category simplify considerably. The first part of the paper is devoted to establishing the main structures of the Fukaya category in the monotone case: the closed-open string maps, weak proper Calabi-Yau structure, Abouzaid's split-generation criterion, and their analogues when weak bounding cochains are included. We then turn to computations of the Fukaya category of the hypersurface $X$: we construct a configuration of monotone Lagrangian spheres in $X$, and compute the associated disc potential. The result coincides with the Hori-Vafa superpotential for the mirror of $X$ (up to a constant shift in the Fano index $1$ case). As a consequence, we give a proof of Kontsevich's homological mirror symmetry conjecture for $X$. We also explain how to extract non-trivial information about Gromov-Witten invariants of $X$ from its Fukaya category.

Simple normal crossing Fano varieties and log Fano manifolds

AbstractA projective log variety (X, D) is called a log Fano manifold if X is smooth and if D is a reduced simple normal crossing divisor on Χ with − (KΧ + D) ample. The n-dimensional log Fano manifolds (X, D) with nonzero D are classified in this article when the log Fano index r of (X, D) satisfies either r ≥ n/2 with ρ(X) ≥ 2 or r ≥ n − 2. This result is a partial generalization of the classification of logarithmic Fano 3-folds by Maeda.

Fano threefolds of large Fano index and large degree

We classify -Fano threefolds of Fano index and sufficiently big degree.Bibliography: 20 titles.

Трехмерные многообразия Фано большого индекса Фано и большой степени

We classify Q-Fano threefolds of Fano index > 2 and big degree.

$\Bbb Q$-Fano threefolds of large Fano index. I.

We study Q-Fano threefolds of large Fano index. In particular, we prove that the maximum possible Fano index is attained only by the weighted projective space P(3,4,5,7) .

Fano 3-folds with divisible anticanonical class

We show the nonvanishing of H0(X,−KX) for any a Fano 3-fold X for which −KX is a multiple of another Weil divisor in Cl(X). The main case we study is Fano 3-folds with Fano index 2: that is, 3-folds X with rank Pic(X)=1, \({\mathbb{Q}}\) -factorial terminal singularities and −KX = 2A for an ample Weil divisor A. We give a first classification of all possible Hilbert series of such polarised varieties (X,A) and deduce both the nonvanishing of H0(X,−KX) and the sharp bound (−KX)3≥ 8/165. We find the families that can be realised in codimension up to 4.

Rationality of an Enriques-Fano threefold of genus five

We prove the rationality of a non-Gorenstein Fano threefold of Fano index one and degree eight having terminal cyclic quotient singularities and Picard group . This threefold can be described as the quotient of a double covering of ramified in a smooth quartic surface by an involution fixing eight different points.

Bounded three-dimensional Fano varieties of integer index

The cube of the anticanonical class of a three-dimensional Fano variety with canonical singularities and integer Fano index is effectively bounded.

Classification of non-Gorenstein Q-Fano d-folds of Fano index greater than d − 2

A d-dimensional normal projective variety X is called a Q-Fano d-fold if it has only terminal singularities and if the anti-canonical Weil divisor – Kx is ample. The singularity index I = I(X) of X is defined to be the smallest positive integer such that – IKX is Cartier. Then there is a positive integer r and a Cartier divisor H such that – IKX ~ rH. Taking the largest number of such r, we call r/I the Fano index of X.

The fundamental group of the smooth part of a log Fano variety

Let X be a normal projective variety over the complex number field C. We call X a Fano variety if X is Q-Gorenstein and the anti-canonical divisor — Kx is ample. A Fano variety X is said to be a log Fano variety if X has only log terminal singularities (cf. [6]). A Fano variety X is called a canonical Fano variety if X has only canonical singularities (cf. [6]). The Cartier index c{X) is the smallest positive integer such that c{X)Kx is a Cartier divisor. The Fano index, denoted by r(X), is the largest positive rational number such that —Kχ~~Q r{X)H (Q-linear equivalence) for a Cartier divisor H. This note consists of two sections. In §1, we shall consider canonical Fano 3-folds and prove the following: