Fano Threefolds Research Articles

Endomorphisms of varieties and Bott vanishing

We show that a projective variety with an int-amplified endomorphism of degree invertible in the base field satisfies Bott vanishing. This is a new way to analyze which varieties have nontrivial endomorphisms. In particular, we extend some classification results on varieties admitting endomorphisms (for Fano threefolds of Picard number one and several other cases) to any characteristic. The classification results in characteristic zero are due to Amerik–Rovinsky–Van de Ven, Hwang–Mok, Paranjape–Srinivas, Beauville, and Shao–Zhong. Our method also bounds the degree of morphisms into a given variety. Finally, we relate endomorphisms to global F F -regularity.

K-stability of Casagrande–Druel varieties

Abstract 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 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 P n − 1 \mathbb{P}^{n-1} ramified over smooth hypersurfaces of degree 2 ⁢ d 2d with n > d > n 2 > 1 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.

Some More Fano Threefolds with a Multiplicative Chow–Künneth Decomposition

We exhibit several families of Fano threefolds with a multiplicative Chow–Künneth decomposition, in the sense of Shen–Vial. As a consequence, a certain tautological subring of the Chow ring of powers of these threefolds injects into cohomology. As a by-product of the argument, we observe that double covers of projective spaces admit a multiplicative Chow–Künneth decomposition.

Key varieties for prime $$\pmb {\mathbb {Q}}$$-Fano threefolds defined by Freudenthal triple systems

Key varieties for prime $$\pmb {\mathbb {Q}}$$-Fano threefolds defined by Freudenthal triple systems

The Manin–Peyre conjecture for smooth spherical Fano threefolds

The Manin–Peyre conjecture is established for smooth spherical Fano threefolds of semisimple rank one and type N. Together with the previously solved case T and the toric cases, this covers all types of smooth spherical Fano threefolds. The case N features a number of structural novelties; most notably, one may lose regularity of the ambient toric variety, the height conditions may contain fractional exponents, and it may be necessary to exclude a thin subset with exceptionally many rational points from the count, as otherwise Manin’s conjecture in its original form would turn out to be incorrect.

On maximally non-factorial nodal Fano threefolds

We classify non-factorial nodal Fano threefolds with 1 node and class group of rank 2 .

K-stability of Fano threefolds of rank 3 and degree 14

K-stability of Fano threefolds of rank 3 and degree 14

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

Automorphisms of Fano threefolds of rank 2 and degree 28

We describe the automorphism groups of smooth Fano threefolds of rank 2 and degree 28 in the cases where they are finite.

On K-stability of P3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {P}^3$$\\end{document} blown up along a quintic elliptic curve

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.

Instanton sheaves on Fano threefolds

Instanton sheaves on Fano threefolds

On K$K$‐stability of P3$\mathbb {P}^3$ blown up along the disjoint union of a twisted cubic curve and a line

On K$K$‐stability of P3$\mathbb {P}^3$ blown up along the disjoint union of a twisted cubic curve and a line

Bott vanishing for Fano threefolds

Bott vanishing for Fano threefolds

Density of rational points on some quadric bundle threefolds

We prove the Manin–Peyre conjecture for the number of rational points of bounded height outside of a thin subset on a family of Fano threefolds of bidegree (1, 2).

Fano Threefolds With Affine Canonical Extensions

Fano Threefolds With Affine Canonical Extensions

On the Futaki invariant of Fano threefolds

We study the zero locus of the Futaki invariant on K-polystable Fano threefolds, seen as a map from the Kähler cone to the dual of the Lie algebra of the reduced automorphism group. We show that, apart from families 3.9, 3.13, 3.19, 3.20, 4.2, 4.4, 4.7 and 5.3 of the Iskovskikh–Mori–Mukai classification of Fano threefolds, the Futaki invariant of such manifolds vanishes identically on their Kähler cone. In all cases, when the Picard rank is greater or equal to two, we exhibit explicit 2-dimensional differentiable families of Kähler classes containing the anti-canonical class and on which the Futaki invariant is identically zero. As a corollary, we deduce the existence of non Kähler–Einstein cscK metrics on all such Fano threefolds.

Footnotes to a footnote of a paper by Fano

Following a suggestion in a footnote of the paper [1] by Fano, we give a direct proof of the fact that one of the two Fano threefolds of degree 72 in P 38 is isomorphic to the anticanonically embedded weighted projective space P ( 1 , 1 , 4 , 6 ) .

Coregularity of smooth Fano threefolds

Coregularity of smooth Fano threefolds

The Manin–Peyre conjecture for smooth spherical Fano varieties of semisimple rank one

Abstract The Manin–Peyre conjecture is established for a class of smooth spherical Fano varieties of semisimple rank one. This includes all smooth spherical Fano threefolds of type T as well as some higher-dimensional smooth spherical Fano varieties.

K-stable smooth Fano threefolds of Picard rank two

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.