Fano Threefolds

We construct families of non-toric ${\mathbb {Q}}$ -factorial terminal Fano ( ${\mathbb {Q}}$ -Fano) threefolds of codimension $\geq 20$ corresponding to 54 mutation classes of rigid maximally mutable Laurent polynomials. From the point of view of mirror symmetry, they are the highest codimension (non-toric) ${\mathbb {Q}}$ -Fano varieties for which we can currently establish the Fano/Landau–Ginzburg correspondence. We construct 46 additional ${\mathbb {Q}}$ -Fano threefolds with codimensions of new examples ranging between 19 and 5. Some of these varieties are presented as toric complete intersections, and others as Pfaffian varieties.

We discuss various results on Hilbert schemes of lines and conics and automorphism groups of smooth Fano threefolds with Picard rank 1. Besides a general review of facts well known to experts, the paper contains some new results, for instance, we give a description of the Hilbert scheme of conics on any smooth Fano threefold of index 1 and genus 10. We also show that the action of the automorphism group of a Fano threefold $X$ of index 2 (respectively, 1) on an irreducible component of its Hilbert scheme of lines (respectively, conics) is faithful if the anticanonical class of $X$ is very ample with a possible exception of several explicit cases. We use these faithfulness results to prove finiteness of the automorphism groups of most Fano threefolds and classify explicitly all Fano threefolds with infinite automorphism group. We also discuss a derived category point of view on the Hilbert schemes of lines and conics, and use this approach to identify some of them.

In this paper we give first examples of $\\mathbb{Q}$-Fano threefolds whose birational Mori fiber structures consist of exactly three $\\mathbb{Q}$-Fano threefolds. These examples are constructed as weighted hypersurfaces in a specific weighted projective space. We also observe that the number of birational Mori fiber structures does not behave upper semi-continuously in a family of $\\mathbb{Q}$-Fano threefolds.

We study Fano threefolds with terminal Gorenstein singularities admitting a `minimal' action of a finite group. We prove that under certain additional assumptions such a variety contains no planes. We also obtain upper bounds for the number of singular points of certain Fano threefolds with terminal factorial singularities.

In this note, we classify all the polarized Fano threefold (X, H) with Bs|H|¬ = ∅. As corollaries we obtained that (1) the very ample part of the conjecture of Fujita holds for smooth Fano threefolds and (2) global Seshadri constants of ample divisors on Fano threefolds are bounded from below by 1 except three types of polarized Fano threefolds.

We prove an optimal Kawamata-Miyaoka-type inequality for terminal $\mathbb Q$-Fano threefolds with Fano index at least $3$. As an application, any terminal $\mathbb Q$-Fano threefold $X$ satisfies the following Kawamata-Miyaoka-type inequality \[ c_1(X)^3 < 3c_2(X)c_1(X). \]Comment: 33 pages, 4 tables. Any comments are welcome. v2: we improve the exposition, 29 pages, 3 tables. v3: Final published vesion

Embedding and partial resolution of complex cones over Fano threefolds

By Jahnke–Peternell–Radloff and Takeuchi, almost Fano threefolds with del Pezzo fibrations were classified. Among them, there exist 10 classes such that the existence of members of these was not proved. In this paper, we construct such examples belonging to each of 10 classes.

We prove that minimal instanton bundles on a Fano threefold $X$ of Picard rank one and index two are semistable objects in the Kuznetsov component $\mathsf{Ku}(X)$, with respect to the stability conditions constructed by Bayer, Lahoz, Macrì and Stellari. When the degree of $X$ is at least 3, we show torsion free generalizations of minimal instantons are also semistable objects. As a result, we describe the moduli space of semistable objects with same numerical classes as minimal instantons in $\mathsf{Ku}(X)$. We also investigate the stability of acyclic extensions of non-minimal instantons.

The purpose of this article is to develop further a method to classify varieties X ⊂ [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="01i" /] having canonical curve section, using Gaussian map computations. In a previous article we applied these techniques to classify prime Fano threefolds, that is Fano threefolds whose Picard group is generated by the hyperplane bundle. In this article we extend this method and classify Fano threefolds of higher index and Mukai varieties , i.e., varieties of dimension four or more with canonical curve sections. First we determine when the Hilbert scheme [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="02i" /] of such varieties X is nonempty. Moreover, in the case of Picard number one, we prove that [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="03i" /] is irreducible and that the examples of Fano-Iskovskih and Mukai form a dense open subset of smooth points of [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="04i" /].

Following the work of Altmann and Hausen we give a combinatorial description for smooth Fano threefolds admitting a 2-torus action. We show that a whole variety of properties and invariants can be read off from this description. As an application we prove and disprove the existence of Kähler-Einstein metrics for some of these Fano threefolds, calculate their Cox rings and some of their toric canonical degenerations.

Building on the work of Nogin, we prove that the braid groupB4acts transitively on full exceptional collections of vector bundles on Fano threefolds withb2= 1 andb3= 0. Equivalently, this group acts transitively on the set of simple helices (considered up to a shift in the derived category) on such a Fano threefold. We also prove that on threefolds withb2= 1 and very ample anticanonical class, every exceptional coherent sheaf is locally free.

In this article, a sequel to "Global Frobenius Liftability I" (math:1708:03777v2), we continue the development of a comprehensive theory of Frobenius liftings modulo $p^2$. We study compatibility of divisors and closed subschemes with Frobenius liftings, Frobenius liftings of blow-ups, descent under quotients by some group actions, stability under base change, and the properties of associated F-splittings. Consequently, we characterise Frobenius liftable surfaces and Fano threefolds, confirming the conjecture stated in our previous paper.

We generalize the classical approach of describing the infinitesimal Torelli map in terms of multiplication in a Jacobi ring to the case of quasi-smooth complete intersections in weighted projective space. As an application, we prove that the infinitesimal Torelli theorem does not hold for hyperelliptic Fano threefolds of Picard rank 1, index 1, degree 4, and study the action of the automorphism group on cohomology. The results of this paper are used to prove Lang-Vojta’s conjecture for the moduli of such Fano threefolds in a follow-up paper.

We define non-ordinary instanton bundles on Fano threefolds $X$ extending the notion of (ordinary) instanton bundles. We determine a lower bound for the quantum number of a non-ordinary instanton bundle, i.e. the degree of its second Chern class, showing the existence of such bundles for each admissible value of the quantum number when $i_X\ge 2$ or $i_X=1$ and $\mathrm{Pic}(X)$ is cyclic. In these cases we deal with the component inside the moduli spaces of simple bundles containing the vector bundles we construct and we study their restriction to lines. Finally we give a monadic description of non-ordinary instanton bundles on $\mathbb{P}^3$ and the smooth quadric studying their loci of jumping lines, when of the expected codimension.