Schedule and Abstracts of Talks

List of talks

Hecke modifications of vector bundles plays a central role in several areas of mathematics. In this talk, we shall introduce the subject and link it with some current research works. In particular, we show the connection with modern number theory (Langlands program) and the Hall algebras theory. In the end, we shall present some recent results concerned with automorphic forms which can be obtained by knowing the Hecke modifications


We shall characterize plane conic–line arrangements with maximal global Tjurina number. The vector bundle of logarithmic vector fields T(-log C) of such curves splits, with splitting type (1,d-2). Such arrangements turn out to include the so called Płoski curves, which can be considered as curves with a second worst singularity, after the worst ones corresponding to concurrent lines. Next we will characterize conic-line arrangements with submaximal global Tjurina number. In some cases the Jacobian scheme turns out to be an eigenscheme of a partially symmetric tensor, and this will allow to give some insight in the corresponding polar map.


Instanton bundles originally appeared in the context of Yang-Mills gauge theory. In their seminal work Atiyah, Drinfeld, Hitchin and Manin established a correspondence between the (anti)self-dual solutions of the Yang-Mills equations on the four-sphere S^4 of topological charge n and the mathematical instantons on its twistor space P^3 namely, the rank 2 holomorphic vector bundles E on P^3 having c1(E)=0, c2(E)=n and such that H^1(E(-2))=0. Both twistor geometry and Yang-Mills theory can be generalized to a 4n-dimensional Quaternion Kahler manifold M; this allows us to defined the notion of instanton bundle on the twistor space Z, a so called contact Fano manifold. In this talk I will recall some features of instantons on contact Fano manifolds and present some properties of these bundles and of their moduli in the case M=G2/SO(4) and Z=G2/U(1)SU(2).


 In this talk I make a (very) small tour on results on vector bundle on projective smooth varieties that are decomposable into a direct sum of line bundles. I focus on the logarithmic bundle of a normal crossing divisor on a smooth variety. In particular I present new results on divisors on a rational normal scroll, investigated in a joint work with Francesco Malaspina.


 In this talk I will report on a recent joint work with M.L. Fania, where we investigate on Ulrich bundles on suitable 3-fold scrolls. More precisely, inspired by previous results of M.L. Fania, M. Lelli Chiesa and J. Pons Llopis, concerning Ulrich bundles of rank one and two on scrolls over Hirzebruch surfaces F_0 and F_1, arising as 3-folds of low degree, we explicitely describe components M(r) of moduli spaces of Ulrich bundles of any rank r>0 on 3-fold scrolls X_e over Hirzebruch surfaces F_e, for any non-negative integer e, showing that such components M(r) are generically smooth, of computed dimension, whose general point is proved to correspond to a slope-stable bundle. The use of some results of V. Antonelli and suitable correspondences shed also some lights for the birational geometry of such components M(r).  The previous results, in particular, allow us to compute the Ulrich complexity of 3-folds X_e and to give an effective proof that they are of Ulrich wild type. 


The singular fibres of the SL(n,C)-Hitchin fibration are described by compactified Prym varieties. For a large class of these singular SL(n,C)-fibres, we provide a derived equivalence with the corresponding PGL(n,C)-Hitchin fibre. Our work is based on an extension of the Fourier-Mukai transform  between compactified Jacobians constructed by Arinkin and Melo-Rapagnetta-Viviani.


Let X be a nonsingular complex projective toric variety of dimension n, equipped with an action of the n-dimensional complex torus T. A coherent torsion-free sheaf E on X is said to be T-equivariant if it admits a lift of the T-action on X, which is linear on the stalks of E. It is known that the tangent bundle T_X of a toric variety X is T-equivariant. In this talk, we will study the problem of \mu-stability of T_X for a toric variety X. This is a joint work with Idranil Biswas, Arijit Dey and Mainak Poddar.


Years ago we gave definitions of ampleness and nefness for Higgs bundles which “feel” the Higgs field. In this talk I will review those notions, describe criteria for ampleness and nefness of Higgs bundles, and show an application to surfaces of general type.


A Higgs bundle consist of an holomorphic bundle over a Riemann surface together with a Higgs field. These objects are in the core of gauge theories, being solutions to a dimensional reduction of Yang-Mills equations, and the geometry of their moduli spaces has been studied extensively for the past decades. Here we will focus on the case of Higgs bundles on non-compact Riemann surfaces, which introduces poles on the Higgs field and even a more richer geometry. In this talk we shall analyse their Poisson structure as well as the integrable systems encoded. We shall not assume any previous knowledge on Higgs bundles, so we shall also provide an overview on the subject. This is based on previous work with Biswas, Martens, Peón-Nieto.


I will start by explaining a beautiful classical construction by Rossi: given F a coherent sheaf on a scheme X, Rossi constructs the blow-up of X along F. I will then use this construction to define reduced Gromov-Witten invariants of complete intersections in all genera. This is work with A. Cobos-Rabano, E. Mann and R. Picciotto.


A distribution D on a projective space P^n is given by a saturated subsheaf TD of the tangent sheaf TP^n. Then TD can be defined as the kernel of the contraction morphism with respect to a homogeneous polynomial differential p-form omega. The vanishing locus of omega is denoted Sing(D), the singular scheme of D. In this talk, we will explore the relationship between TD and Sing(D). In particular, we will see how to use syzygies to construct distributions with a prescribed singular scheme. 


I will talk about the geometry of the global nilpotent cone focusing on the role of very stable and wobbly Higgs bundles. I will review the definitions and give some context motivating the interest of these objects. Finally,  I will discuss the classification of nilpotent order two fixed point components in terms of wobbliness and very stability, and  illustrate it via the rank three case. 


Let R=k[x,y,z]. A reduced plane curve C=V(f) in P^2 is free if its associated module of tangent derivations Der(f) is a free R-module, or equivalently if the corresponding sheaf T(-log C) of vector fields tangent to C splits as a direct sum of line bundles on P^2. In general, free curves are difficult to find, and in this note, we describe a new method for constructing free curves in P^2. The key tool in our approach are eigenschemes and pencil of curves, combined with an interpretation of Saito's criterion in this context. Previous constructions typically applied only to curves with quasihomogeneous singularities, which is not necessary in our approach. We illustrate our method by constructing large families of free curves.


On a smooth projective curve the locus of stable bundles that remain stable on all etale Galois covers prime to the characteristic defines a big open in the moduli space of stable bundles. In particular, the bundles trivialized on some etale Galois cover of degree prime to the characteristic are not dense - in contrast to a theorem of Ducrohet and Mehta stating that all etale trivializable bundles are dense in positive characteristic. As an application we study the closure of the prime to p etale trivializable vector bundles. This closure is closely related to a stratification of the moduli space of stable vector bundles via their decomposition behaviour on Galois covers of degree prime to the characterstic. We obtain mostly sharp dimension estimates for the closure of the prime to p etale trivializable bundles as well as the decomposition strata