Daniele Faenzi

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.

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.

Given a smooth genus two curve C C , the moduli space S U C ( 3 ) SU_C(3) of rank three semi-stable vector bundles on C C with trivial determinant is a double cover in P 8 {\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 C . In this paper we continue our exploration of the connections of such moduli spaces with the representation theory of G L 9 GL_9 , initiated by Gruson, Sam and Weyman. In this spirit, starting from a general trivector v v in ∧ 3 C 9 \wedge ^3{\mathbb {C}}^9 , we construct a Fano manifold D Z 10 ( v ) D_{Z_{10}}(v) in G ( 3 , 9 ) G(3,9) as a so-called orbital degeneracy locus, and we prove that it defines a family of Hecke lines in S U C ( 3 ) SU_C(3) . We deduce that D Z 10 ( v ) D_{Z_{10}}(v) is isomorphic to the odd moduli space S U C ( 3 , O C ( c ) ) SU_C(3, {\mathcal {O}}_C(c)) of rank three stable vector bundles on C C with fixed effective determinant of degree one. We deduce that the intersection of D Z 10 ( v ) D_{Z_{10}}(v) with a general translate of G ( 3 , 7 ) G(3,7) in G ( 3 , 9 ) G(3,9) is a K3 surface of genus nineteen.

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.

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 Miró‐Roig.

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.

Given a smooth genus three curve C, the moduli space of rank two stable vector bundles on C with trivial determinant embeds in .

Let k be a field of characteristic zero and R = k[x 1 ,. .. , x n ] be a graded commutative ring. To any set of 1 ≤ s ≤ n-1 algebraically independent polynomials f 1 ,. .. , f s ∈ R one associates a rank-(n+1-s) R-module Der(f 1 ,. .. , f s) of logarithmic derivations; these logarithmic derivations δ ∈ Der(f 1 ,. .. , f s) verify δ(f 1) f 1 = • • • = δ(f s) f s ∈ R. The aim of this note is to extend the Saito criterion, which characterizes freeness of the module Der(f) of tangent vector fields along a reduced divisor V (f), to the module Der(f 1 ,. .. , f s). More precisely, let δ i be (n + 1-s) independent derivations in Der(f 1 ,. .. , f s) and U ⊂ Der(f 1 ,. .. , f s) the sub-module generated by the δ i. We prove that U = Der(f 1 ,. .. , f s) if and only if codim k n V (n+1-s U) > 1 when s > 1 or V (n U) = V (f 1) when s = 1.