Fano 3-folds Research Articles

One-dimensional components in the K-moduli of smooth Fano 3-folds

By identifying K-polystable limits in 4 specific deformations families of smooth Fano 3-folds, we complete the classification of one-dimensional components in the K-moduli space of smoothable Fano 3-folds.

Combinatorial and algebraic mutations of toric Fano 3-folds and mass deformations of 2d(0,2) quiver gauge theories

We argue that algebraic and combinatorial polytope mutations of Fano 3-folds can be identified with mass deformations of associated 2d (0,2) supersymmetric gauge theories realized by brane brick models. These are type IIA brane configurations that realize a large family of 2d worldvolume theories on probe D1-branes at toric Calabi-Yau 4-folds. We show that brane brick models that are related by mass deformations associated to algebraic and combinatorial polytope mutations of Fano 3-folds have mesonic moduli spaces with the same number of generators. We show that mesonic flavor charges of these generators form convex reflexive lattice polytopes that are dual to the toric diagrams of the Fano 3-folds. The generating function of mesonic gauge invariant operators, also known as the Hilbert series of the mesonic moduli space, appears to be identical for such brane brick models under a particular refinement originating from the U(1)R charges in the brane brick model following the mass deformation. Published by the American Physical Society 2024

BIRATIONAL GEOMETRY OF SEXTIC DOUBLE SOLIDS WITH A COMPOUND SINGULARITY

Abstract Sextic double solids, double covers of $\mathbb P^3$ branched along a sextic surface, are the lowest degree Gorenstein terminal Fano 3-folds, hence are expected to behave very rigidly in terms of birational geometry. Smooth sextic double solids, and those which are $\mathbb Q$ -factorial with ordinary double points, are known to be birationally rigid. In this paper, we study sextic double solids with an isolated compound $A_n$ singularity. We prove a sharp bound $n \leq 8$ , describe models for each n explicitly, and prove that sextic double solids with $n> 3$ are birationally nonrigid.

K-stability of Fano 3-folds of Picard rank 3 and degree 22

AbstractWe prove K-stability of smooth Fano 3-folds of Picard rank 3 and degree 22 that satisfy very explicit generality condition.

2 n2-inequality for cA1 points and applications to birational rigidity

The $4 n^2$ -inequality for smooth points plays an important role in the proofs of birational (super)rigidity. The main aim of this paper is to generalize such an inequality to terminal singular points of type $cA_1$ , and obtain a $2 n^2$ -inequality for $cA_1$ points. As applications, we prove birational (super)rigidity of sextic double solids, many other prime Fano 3-fold weighted complete intersections, and del Pezzo fibrations of degree $1$ over $\mathbb {P}^1$ satisfying the $K^2$ -condition, all of which have at most terminal $cA_1$ singularities and terminal quotient singularities. These give first examples of birationally (super)rigid Fano 3-folds and del Pezzo fibrations admitting a $cA_1$ point which is not an ordinary double point.

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).

K-stability of Fano 3-folds of Picard rank 3 and degree 20

We prove K-stability of smooth Fano 3-folds of Picard rank 3 and degree 20 that satisfy very explicit generality condition.

Fano 4-folds with b2>12\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$b_{2}>12$\\end{document} are products of surfaces

Let X\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$X$\\end{document} be a smooth, complex Fano 4-fold, and ρX\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\rho _{X}$\\end{document} its Picard number. We show that if ρX>12\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\rho _{X}>12$\\end{document}, then X\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$X$\\end{document} is a product of del Pezzo surfaces. The proof relies on a careful study of divisorial elementary contractions f:X→Y\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$f\\colon X\ o Y$\\end{document} such that dimf(Exc(f))=2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\dim f(\\operatorname{Exc}(f))=2$\\end{document}, together with the author’s previous work on Fano 4-folds. In particular, given f:X→Y\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$f\\colon X\ o Y$\\end{document} as above, under suitable assumptions we show that S:=f(Exc(f))\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$S:=f(\\operatorname{Exc}(f))$\\end{document} is a smooth del Pezzo surface with −KS=(−KY)|S\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$-K_{S}=(-K_{Y})_{|S}$\\end{document}.

Fano 4-folds with nef tangent bundle in positive characteristic

In characteristic 0, the Campana-Peternell conjecture claims that the only smooth Fano variety with nef tangent bundle should be homogeneous. In this paper, we study the positive characteristic version of the Campana-Peternell conjecture. In particular, we give an affirmative answer for Fano 4-folds with nef tangent bundle and Picard number greater than one.

Correspondence of Donaldson-Thomas and Gopakumar-Vafa invariants on local Calabi-Yau 4-folds over V5 and V22

We compute Gromov-Witten (GW) and Donaldson-Thomas (DT) invariants (and also descendant invariants) for local CY 4-folds over Fano 3-folds, V5 and V22 up to degree 3. We use torus localization for GW invariants computation, and use classical results for Hilbert schemes on V5 and V22 for DT invariants computation. From these computations, one can check correspondence between DT and Gopakumar-Vafa (GV) invariants conjectured by Cao-Maulik-Toda in genus 0. Also we can compute genus 1 GV invariants via the conjecture of Cao-Toda, which turned out to be 0. These fit into the fact that there are no smooth elliptic curves in V5 and V22 up to degree 3.

Fano 3-folds and double covers by half elephants

Fano 3-folds and double covers by half elephants

Polyvector fields for Fano 3-folds

We compute the Hochschild–Kostant–Rosenberg decomposition of the Hochschild cohomology of Fano 3-folds. This is the first step in understanding the non-trivial Gerstenhaber algebra structure of this invariant, and yields some initial insights in the classification of Poisson structures on Fano 3-folds of higher Picard rank.

Nonsolidity of uniruled varieties

AbstractWe give conditions for a uniruled variety of dimension at least 2 to be nonsolid. This study provides further evidence to a conjecture by Abban and Okada on the solidity of Fano 3-folds. To complement our results we write explicit birational links from Fano 3-folds of high codimension embedded in weighted projective spaces.

The Lefschetz defect of Fano varieties

Smooth, complex Fano varieties are a very natural class of projective varieties, and have been intensively studied both classically and in recent times, due to their significance in the framework of the Minimal Model Program. The Lefschetz defect is an invariant of smooth Fano varieties that has been recently introduced in [7]; it is related to the Picard number \(\rho _X\) of X, and to the Picard number of prime divisors in X. This paper is a survey on this new invariant and its properties: we explain the definition of the Lefschetz defect \(\delta _X\) and its origin, the known results, and several examples, especially in dimensions 3 and 4; we do not give new theoretical results. In particular, we explain how the results on the Lefschetz defect allow to recover the classification of Fano 3-folds with \(\rho _X\ge 5\), due to Mori and Mukai. We also review the known examples of Fano 4-folds with \(\rho _X\ge 6\), that are remarkably few: apart products and toric examples, there are only 6 known families, with \(\rho _X\in \{6,7,8,9\}\).

Universal Structures in C-Linear Enumerative Invariant Theories

An enumerative invariant theory in Algebraic Geometry, Differential Geometry, or Representation Theory, is the study of invariants which 'count' $\tau$-(semi)stable objects $E$ with fixed topological invariants $[E]=\alpha$ in some geometric problem, using a virtual class $[{\cal M}_\alpha^{\rm ss}(\tau)]_{\rm virt}$ in some homology theory for the moduli spaces ${\cal M}_\alpha^{\rm st}(\tau)\subseteq{\cal M}_\alpha^{\rm ss}(\tau)$ of $\tau$-(semi)stable objects. Examples include Mochizuki's invariants counting coherent sheaves on surfaces, Donaldson-Thomas type invariants counting coherent sheaves on Calabi-Yau 3- and 4-folds and Fano 3-folds, and Donaldson invariants of 4-manifolds. We make conjectures on new universal structures common to many enumerative invariant theories. Such theories have two moduli spaces ${\cal M},{\cal M}^{\rm pl}$, where the second author gives $H_*({\cal M})$ the structure of a graded vertex algebra, and $H_*({\cal M}^{\rm pl})$ a graded Lie algebra, closely related to $H_*({\cal M})$. The virtual classes $[{\cal M}_\alpha^{\rm ss}(\tau)]_{\rm virt}$ take values in $H_*({\cal M}^{\rm pl})$. Defining $[{\cal M}_\alpha^{\rm ss}(\tau)]_{\rm virt}$ when ${\cal M}_\alpha^{\rm st}(\tau)\ne{\cal M}_\alpha^{\rm ss}(\tau)$ (in gauge theory, when the moduli space contains reducibles) is a difficult problem. We conjecture that there is a natural way to define $[{\cal M}_\alpha^{\rm ss}(\tau)]_{\rm virt}$ in homology over $\mathbb Q$, and that the resulting classes satisfy a universal wall-crossing formula under change of stability condition $\tau$, written using the Lie bracket on $H_*({\cal M}^{\rm pl})$. We prove our conjectures for moduli spaces of representations of quivers without oriented cycles. Our conjectures in Algebraic Geometry using Behrend-Fantechi virtual classes are proved in the sequel arXiv:2111.04694.

Fano 3-folds, reflexive polytopes and brane brick models

Reflexive polytopes in n dimensions have attracted much attention both in mathematics and theoretical physics due to their connection to Fano n-folds and mirror symmetry. This work focuses on the 18 regular reflexive polytopes corresponding to smooth Fano 3-folds. For the first time, we show that all 18 regular reflexive polytopes have corresponding 2d (0, 2) gauge theories realized by brane brick models. These 2d gauge theories can be considered as the worldvolume theories of D1-branes probing the toric Calabi-Yau 4-singularities whose toric diagrams are given by the associated regular reflexive polytopes. The generators of the mesonic moduli space of the brane brick models are shown to form a lattice of generators due to the charges under the rank 3 mesonic flavor symmetry. It is shown that the lattice of generators is the exact polar dual reflexive polytope to the corresponding toric diagram of the brane brick model. This duality not only highlights the close relationship between the geometry and 2d gauge theory, but also opens up pathways towards new discoveries in relation to reflexive polytopes and brane brick models.

Fano 4-folds with a small contraction

Let X be a smooth complex Fano 4-fold. We show that if X has a small elementary contraction, then ρX≤12, where ρX is the Picard number of X. This result is based on a careful study of the geometry of X, on which we give a lot of information. We also show that in the boundary case ρX=12 an open subset of X has a smooth fibration with fiber P1. Together with previous results, this implies that if X is a Fano 4-fold with ρX≥13, then every elementary contraction of X is divisorial and sends a divisor to a surface. The proof is based on birational geometry and the study of families of rational curves. More precisely the main tools are: the study of families of lines in Fano 4-folds and the construction of divisors covered by lines, a detailed study of fixed prime divisors, the properties of the faces of the effective cone, and a detailed study of rational contractions of fiber type.

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].

Terminal toric Fano 3-folds with numerical conditions

We completely classify the Q-factorial terminal toric Fano three-folds such that the sum of the squared torus invariant prime divisors is non-negative.

Tom and Jerry triples with an application to Fano 3-folds

Unprojection is a theory due to Reid which constructs more complicated rings starting from simpler data. In the present work, we develop a new method of unprojection. Starting from a codimension 3 ideal defined by the Pfaffians of a 5 × 5 skewsymmetric matrix, we describe a number of unprojection formats that construct Gorenstein rings of codimension 6. We give two applications that construct families of Fano 3-folds embedded anticanonically in codimension 6 that existing methods do not realize.