arXiv subjects

Daniele Faenzi

Publications and source records attributed to Daniele Faenzi.

At least 19 recordsLinked to original sources

arXiv2026

Higher-rank instantons sheaves on Fano threefolds

We define instanton sheaves of higher rank on smooth Fano threefolds X of Picard rank one and show that their topological classification depends on two integers, namely the rank n (or the half of it, if the Fano index of X is odd) and the charge k. We elucidate the value of the minimal charge k0 of slope-stable n-instanton bundles (except for Fano threefolds of index 1 and genus 3 or 4), as an integer depending only on the genus of X and on n and we prove the existence of slope-stable n-instanton bundles of charge k greater than k0. Next, we study the acyclic extension of instantons on Fano threefolds with curvilinear Kuznetsov component and give a monadic description when the intermediate Jacobian is trivial. Finally, we provide several features of a general element in the main component of the moduli space of intantons, such as and generic splitting over rational curves contained in X and stable restriction to a K3 section S of X, and give applications to Lagrangian subvarieties of moduli spaces of sheaves on S.

math.AG↗

arXiv2026

Logarithmic Derivations of Adjoint Discriminants

We exhibit a relationship between projective duality and the sheaf of logarithmic vector fields along a reduced divisor $D$ of projective space, in that the push-forward of the ideal sheaf of the conormal variety in the point-hyperplane incidence, twisted by the tautological ample line bundle is isomorphic to logarithmic differentials along $D$. Then we focus on the adjoint discriminant $D$ of a simple Lie group with Lie algebra $\mathfrak{g}$ over an algebraically closed field $\mathbf{k}$ of characteristic zero and study the logarithmic module $\mathrm{Der}_{\mathbf{U}}(-\log(D))$ over $\mathbf{U} = \mathbf{k}[\mathfrak{g}]$. When $\mathfrak{g}$ is simply laced, we show that this module has two direct summands: the $G$-invariant part, which is free with generators in degrees equal to the exponents of $G$, and the $G$-variant part, which is of projective dimension one, presented by the Jacobian matrix of the basic invariants of $G$ and isomorphic to the image of the map $\mathbf{ad}\,: \mathfrak{g} \otimes \mathbf{U}(-1) \rightarrow \mathfrak{g} \otimes \mathbf{U}$ given by the Lie bracket. When $\mathfrak{g}$ is not simply laced, we give a length-one equivariant graded free resolution of $\mathrm{Der}_{\mathbf{U}}(-\log(D))$ in terms of the exponents of $G$ and of the quasi-minuscule representation of $G$.

math.AG↗

arXiv2026

On the projective dimension of some deformations of Weyl arrangements

We show that the logarithmic derivation module of (the cone of) the deformation A of a Weyl arrangement associated with a root system of simply laced type has projective dimension one if the deforming parameter ranges from -j to j+2. In addition, we give an explicit minimal free resolution when the root system is of type A3 and B2. Moreover, in the second case, we determine the jumping lines of maximal jumping order of the associated vector bundle. When the deforming parameter of A (respectively A') ranges from -k to k+j (respectively, from -k' to k'+j), with k different from k' and j at least 3, this allows to distinguish D0(A) from D0(A') shifted by 4(k'-k), even though these modules have the same graded Betti numbers.

math.AG↗

arXiv2026

Logarithmic sheaves of complete intersections

We define logarithmic tangent sheaves associated with complete intersections in connection with Jacobian syzygies and distributions. We analyse the notions of local freeness, freeness and stability of these sheaves. We carry out a complete study of logarithmic sheaves associated with pencils of quadrics and compute their projective dimension from the classical invariants such as the Segre symbol and new invariants (splitting type and degree vector) designed for the classification of irregular pencils. This leads to a complete classification of free (equivalently, locally free) pencils of quadrics. Finally we produce examples of locally free, non free pencils of surfaces in P3 of any degree k at least 3, answering (in the negative) a question of Calvo-Andrade, Cerveau, Giraldo and Lins Neto about codimension foliations on P3 .

math.AG↗

arXiv2025

Ulrich ranks of Veronese varieties and equivariant instantons

We construct Ulrich bundles on Veronese threefolds of arbitrary degree as generic deformations of symmetric squares of equivariant instanton bundles on the projective space, thus classifying the rank of Ulrich bundles on such varieties and proving a conjecture of Costa and Mir{ó}-Roig.

math.AC↗

arXiv2025

Ulrich and Instanton Bundles on Special Cubic Fourfolds

We study instanton and Ulrich bundles on hypersurfaces of the projective space, with a focus on special cubic fourfolds and generalized Pfaffians, notably defined by skew-symmetric endomorphisms of Steiner bundles. We prove that the acyclic extensions of instantons deform to Ulrich bundles and deduce that the existence of instantons of low rank and charge implies the existence of Ulrich bundles of low rank, which in turn forces the fourfold to lie in some Hassett divisor. Finally we take a closer look to divisors of cubics with discriminant 18 and 20.

math.AG↗

arXiv2025

Anti-symplectic involutions on moduli spaces of sheaves on K3 surfaces via auto-equivalences

We provide new examples of anti-symplectic involutions on moduli spaces of stable sheaves on K3 surfaces. These involutions are constructed through (anti) autoequivalences of the bounded derived category of coherent sheaves on K3 surfaces arising from spherical bundles. We analyze these induced maps in the moduli space, imposing restrictions on the Mukai vector and considering the preservation of stability conditions. Our construction extends and unifies classical examples, such as the Beauville involutions, Markman-O'Grady reflections and a more recent construction by Beri-Manivel.

math.AG↗

arXiv2025

Göpel Varieties

We show that the Coble hypersurfaces, uniquely characterized by the remarkable property that their singular loci are an abelian surface and a Kummer threefold, respectively, belong to a family of hypersurfaces exhibiting similar behavior, but defined in various types of homogeneous spaces. With the help of Jordan-Vinberg theory, we show how these hypersurfaces can be parametrized by G{ö}pel type varieties inside projectivized representations of complex reflection groups.

math.AG↗

arXiv2025

Equivariant spaces of matrices of constant corank one

We study spaces of matrices coming from irreducible representations of reductive groups over an algebraically closed field of characteristic zero and we completely classify those of constant corank one. In particular, we recover the examples coming from symmetric forms discovered in [BFL22].

math.AG↗

arXiv2024

Logarithmic vector fields and foliations on toric varieties

We introduce a toric version of the sheaf of logarithmic vector fields along a divisor of a simplicial toric variety. The notion is also relevant for algebraically independent families of polynomials in the Cox ring. We provide a generalisation of the Saito criterion for the freeness of the toric logarithmic sheaf. We explain the relationship between this sheaf and the usual sheaf of logarithmic vector fields and the connection with holomorphic foliations on toric varieties.

math.AG↗

arXiv2024

A generalized Saito freeness criterion

We establish generalizations of Saito's criterion for the freeness of divisors in projective spaces that apply both to sequences of several homogeneous polynomials and to divisors on other complete varieties. As an application, the new criterion is applied to several examples, including sequences whose polynomials depend on disjoint sets of variables, some sequences that are equivariant for the action of a linear group, blow-ups of divisors, and certain sequences of polynomials in positive characteristics.

math.AC↗

arXiv2024

Saito criterion and its avatars

Saito gave a nice and efficient criterion to determine whether the module of logarithmic derivation associated with a reduced divisor in a complex variety is free or not. The aim of this note is to propose a new proof of this criterion, in the affine space, in the projective space, and for multiderivations, based on straightforward observations concerning free and reflexive modules. This point of view also allows us to prove a generalized version of the Saito criterion that applies to derivation modules associated with several polynomials.

math.AG↗

arXiv2023

Hecke cycles on moduli of vector bundles and orbital degeneracy loci

Given a smooth genus two curve $C$, the moduli space SU$_C(3)$ of rank three semi-stable vector bundles on $C$ with trivial determinant is a double cover in $\mathbb{P}^8$ branched over a sextic hypersurface, whose projective dual is the famous Coble cubic, the unique cubic hypersurface that is singular along the Jacobian of $C$. In this paper we continue our exploration of the connections of such moduli spaces with the representation theory of $GL_9$, initiated in \cite{GSW} and pursued in \cite{GS, sam-rains1, sam-rains2, bmt}. Starting from a general trivector $v$ in $\wedge^3\mathbb{C}^9$, we construct a Fano manifold $D_{Z_{10}}(v)$ in $G(3,9)$ as a so-called orbital degeneracy locus, and we prove that it defines a family of Hecke lines in SU$_C(3)$. We deduce that $D_{Z_{10}}(v)$ is isomorphic to the odd moduli space SU$_C(3, \mathcal{O}_C(c))$ of rank three stable vector bundles on $C$ with fixed effective determinant of degree one. We deduce that the intersection of $D_{Z_{10}}(v)$ with a general translate of $G(3,7)$ in $G(3,9)$ is a K3 surface of genus $19$.

math.AG↗

arXiv2023

Derived category of the spinor 15-fold

We construct a full exceptional Lefschetz collection on the spinor 15-fold consisting of a connected component of the space of orthogonal 6-dimensional subspaces of a 12-dimensional complex vector space, isotropic with respect of a fixed non-degenerate quadratic form. The collection is made of 2 twists of a 4-item block and 8 twists of a 3-item block, confirming a conjecture of Kuznetsov and Smirnov. We speculate that a similar collection might work for the Freudenthal E7-variety.

math.AG↗

arXiv2023

Rationality of peskine varieties

We study the rationality of the Peskine sixfolds in P^9. We prove the rationality of the Peskine sixfolds in the divisor D^{3,3,10} inside the moduli space of Peskine sixfolds and we provide a cohomological condition which ensures the rationality of the Peskine sixfolds in the divisor D^{1,6,10} (notation from [BS]). We conjecture, as in the case of cubic fourfolds containing a plane, that the cohomological condition translates into a cohomological and geometric condition involving the Debarre-Voisin hyperk{ä}hler fourfold associated to the Peskine sixfold.

math.AG↗

arXiv2023

The Coble Quadric

Given a smooth genus three curve $C$, the moduli space of rank two stable vector bundles on C with trivial determinant embeds in $\mathbb{P}^8$ as a hypersurface whose singular locus is the Kummer threefold of $C$; this hypersurface is the Coble quartic. Gruson, Sam and Weyman realized that this quartic could be constructed from a general skew-symmetric fourform in eight variables. Using the lines contained in the quartic, we prove that a similar construction allows to recover SU$_C(2, L)$, the moduli space of rank two stable vector bundles on C with fixed determinant of odd degree L, as a subvariety of $G(2, 8)$. In fact, each point $p \in C$ defines a natural embedding of SU$_C(2, \mathcal{O}(p))$ in $G(2, 8)$. We show that, for the generic such embedding, there exists a unique quadratic section of the Grassmannian which is singular exactly along the image of SU$_C(2, \mathcal{O}(p))$, and thus deserves to be coined the Coble quadric of the pointed curve $(C, p)$.

math.AG↗

arXiv2023

Codimension one foliations on homogeneous varieties

The aim of this paper is to study codimension one foliations on rational homogeneous spaces, with a focus on the moduli space of foliations of low degree on Grassmannians and cominuscule spaces. Using equivariant techniques, we show that codimension one degree zero foliations on (ordinary, orthogonal, symplectic) Grassmannians of lines, some spinor varieties, some Lagrangian Grassmannians, the Cayley plane (an $E_6$-variety) and the Freudenthal variety (an $E_7$-variety) are identified with restrictions of foliations on the ambient projective space. We also provide some evidence that such results can be extended beyond these cases.

math.AG↗

arXiv2022

Ulrich bundles on cubic fourfolds

We show the existence of rank 6 Ulrich bundles on a smooth cubic fourfold. First, we construct a simple sheaf E of rank 6 as an elementary modification of an ACM bundle of rank 6 on a smooth cubic fourfold. Such an E appears as an extension of two Lehn-Lehn-Sorger-van Straten sheaves. Then we prove that a general deformation of E(1) becomes Ulrich. In particular, this says that general cubic fourfolds have Ulrich complexity 6.

math.AG↗