Titles and Abstracts

Cristian Anghel (IMAR): Globally generated vector bundles on $\mathbb{P}^3$ and the unirationality of $\mathcal{M}_g$, $g \leq 13$.

Abstract: We combine the idea of Chang and Ran [Invent. Math. 76 (1984), 41--54] of using monads of vector bundles on the projective 3-space to prove the unirationality of the moduli spaces of curves of low genus with our classification of globally generated vector bundles with the first Chern class $c_1 = 5$ on the projective 3-space [arXiv:1805.11336] to get an alternative argument for the unirationality of the moduli spaces of curves of degree at most 13 (following the general guidelines of the method of Chang and Ran but with quite different effective details). This is a joint work with I. Coanda and N. Manolache.

Marian Aprodu (University of Bucharest): Koszul modules and applications

Abstract: Koszul modules were introduced in 2015 by Ş. Papadima and A. Suciu in a topological setup. They represent infinitesimal versions of Alexander invariants. I will report on some recent joint works with G. Farkas, Ş. Papadima, C. Raicu, A. Suciu and J. Weyman on several applications of these objects in algebraic geometry and geometric group theory.

Ada Boralevi (Politecnico di Torino): Matrices of constant rank, quadric surfaces, Pfaffian hypersurfaces

Abstract: I will report on a recent joint work with M.L. Fania and E. Mezzetti, where we studied smooth quadric surfaces in the Pfaffian hypersurface in P^14, not intersecting the Grassmannian G(1,5). Such surfaces correspond to quadratic systems of skew-symmetric matrices of size 6 and constant rank 4, and give rise to globally generated vector bundles on the quadric. I will describe these bundles and their geometry, explaining the different techniques that we used.

Gaia Comaschi (Campinas): Instanton sheaves of low charge on Fano threefolds

Abstract: Let $X$ be a Fano threefold of Picard number one and of index $2+h, \ h=0,1$. An \textit{instanton sheaf of charge $k$ on $X$} is defined as a semi-stable rank 2 torsion free sheaf $F$ with Chern classes $c_1=-h, \ c_2=k, \ c_3=0$ and such that $F(-1)$ has no cohomology. Locally free instantons, originally defined on the projective space and later generalised on other Fano threefolds $X$, had been largely studied from several authors in the past years; their moduli spaces present an extremely rich geometry and useful applications to the study of curves on $X$. In this talk I will illustrate several features of non-locally free instantons of low charge on 3 dimensional quadrics and cubics. I will focus in particular on the role that they play in the study of the Gieseker-Maruyama moduli space $M_X(2;-h,k,0)$ and describe how we can still relate these sheaves to curves on $X$.

Daniele Faenzi (IMB): Logarithmic sheaves of complete intersections

Abstract: We define logarithmic tangent sheaves associated with regular sequences and study some of their properties, like (local) freeness and stability. A special focus will be on pencils, most notably in low degree or dimension. Based on joint work with M. Jardim and J. Vallès.

Soheyla Feyzbaksh (Imperial College London): Hyperkähler varieties as Brill-Noether loci on curves

Abstract: Consider the moduli space $M_C(r; K_C)$ of stable rank r vector bundles on a curve $C$ with canonical determinant, and let $h$ be the maximum number of linearly independent global sections of these bundles. If $C$ embeds in a K3 surface $X$ as a generator of $Pic(X)$ and the genus of $C$ is sufficiently high, I will show the Brill-Noether locus $\BN_C \subset M_C(r; K_C)$ of bundles with $h$ global sections is a smooth projective Hyperkähler manifold, isomorphic to a moduli space of stable vector bundles on $X$. The main technique is to apply wall-crossing with respect to Bridgeland stability conditions on K3 surfaces.

Abdelmoubine Amar Henni (Federal University of Santa Catarina): Analogue of t’Hooft Instantões on the blow up of projective space

Abstract: We define the analogue of instanton sheaves on the blow-up ℙn˜ of the n−dimensional projective space at a point. We choose appropriate polarisation on ℙn˜ and construct rank 2 examples of locally free and non locally free (but torsion free) type. In general, the defined instantons also turn out to be cohomology of monads, although non linear ones. Moreover, in the five dimensional case, we show that there are continuous families of them that fill, at least, a smooth component in the moduli of semi-stable sheaves.

Angelo Felice Lopez (Università degli Studi Roma Tre): Positivity of Ulrich vector bundles

Abstract: An Ulrich vector bundle is globally generated, but how much is it positive? In the first part of the talk we will outline a recent result that allows to characterize ample Ulrich vector bundles only in terms of their restriction to lines contained in the variety X. On the other hand, if an Ulrich bundle is not big, then X is covered by a family of disjoint linear spaces and this allows, in many cases, to classify non-big Ulrich vector bundles. We will give an account of recent results and conjectures in this direction. Work in collaboration with José Carlos Sierra Garcia and Roberto Muñoz.

Mario Maican (IMAR): Variation of moduli spaces of coherent systems of dimension one and order one

Abstract: We study the wall-crossing for moduli spaces of coherent systems of dimension one and order one on a smooth projective variety over the complex numbers. We compute the topological Euler characteristic of the moduli spaces in the particular case when the variety is a quadric surface and the first Chern class of the coherent systems is of the form (2, r).

Simone Marchesi (Universitat de Barcelona): Logarithmic sheaves with fixed points

Abstract: In this talk we generalize the notion of logarithmic tangent sheaf, considering a fixed set of points Z of a divisor D. In particular, studying particular smooth surfaces, we will determine when the pair (D,Z) can be recovered from the generalized logarithmic sheaf, a property that is referred to as Torelli property. This is a joint work with S. Huh, J. Pons-Llopis and J. Vallès.

Dimitri Markushevich (Université de Lille): A case study in complex crystallographic groups: point group SL(2,7)

Abstract: A complex crystallographic (CC) group Γ is a discrete group of affine transformations of the complex space C^n acting with compact quotient. Any such group is an extension of a finite linear group G, called the point group, by a lattice L of maximal rank 2n. A CC group is of reflection type (a CCR group) if it is generated by affine reflections. A conjecture of Bernstein-Schwarzman suggests that the quotient C^n/Γ is a weighted projective space when Γ is irreducible; this is a natural generalization of Shephard-Todd-Chevalley theorem for finite linear groups generated by reflections. The conjecture is known in dimension 2 and for CCR groups of Coxeter type, that is those whose point group G is conjugate to a real Coxeter group. In the talk the case of a genuinely complex CCR group Γ in dimension 3 will be discussed, with quasi-simple point group G of order 336. In this case C^n/Γ can be interpreted as the quotient of the Jacobian of Klein's quartic curve by its full automorphism group {±1}×H, where H is Klein's simple group of order 168. This is a joint work with Anne Moreau.

Cristian Martinez (Campinas): Vertical asymptotics for stability conditions on threefolds

Abstract: Let $(X,H)$ be a smooth, polarized, projective variety. One possible first step, in order to use the power of derived categories to study the moduli space of $H$-Gieseker semistable sheaves, is to be able to put Gieseker stability in the language of stability conditions. While on surfaces this can be easily achieve (by taking the volume $H^2$ to be sufficiently large), on threefolds the Gieseker condition is naturally only asymptotic. In fact, as proven by Jardim and Maciocia and by Pretti, Gieseker stability on threefolds can be interpreted as a limiting notion of Bridgeland stability.

Now, suppose that $X$ is a threefold where the conjectural construction of stability conditions due to Bayer, Macr\`i and Toda works. If we were to attempt to extend the wall-crossing techniques used on surfaces to these cases, then the so called vertical asymptotics are the perfect analog of the surface situation: we work with a 1-parameter family of tilts of a fixed category, and for each of such tilts a 1-parameter family of Bridgeland slopes.

In this talk, we will characterize the limit semistable objects at large volume in complete generality and provide some examples of these asymptotically semistable objects. When $X=\mathbb{P}^3$ we will study in detail the case $\mathrm{ch}(E)=(-R,0,D,0)$, which is a perfect analog of $\mathbb{P}^2$: there is a 2-dimensional slice of stability conditions with a finite number of nested walls whose innermost chamber lies inside a quiver region, and when $R=0$ the only semistable objects in the outermost chamber are the 1-dimensional Gieseker semistable sheaves. For all values $R\geq 0$, the only limit semistable objects of the form $E$ or $E[1]$ (where $E$ is a sheaf) that do not get destabilized until the innermost wall are precisely the (shifts of) instanton sheaves. This is joint work with Marcos Jardim and Antony Maciocia.

Giorgio Ottaviani (Università degli Studi di Firenze): The Hessian map

Abstract: The Hessian map $h_{d,r}$ associates to any hypersurface of degree $d\ge 3$ in $\mathbb{P}^r$ its Hessian hypersurface. It is a rational map

h_{d,r}\colon\mathbb{P}\left(\mathrm{Sym}^d\C^{r+1}\right)\dashrightarrow\P\left(\sym^{(r+1)(d-2)}\C^{r+1}\right).

There are three classical cases where $h_{d,r}$ is not birational, namely

$h_{3,1}$, which cannot be birational by dimensional reasons and indeed it is a $\P^1$-fibration,
$h_{4,1}$ which is $2\colon 1$,
$h_{3,2}$ which is $3\colon 1$.

In a paper joint with Ciro Ciliberto, we conjecture that these are the only three cases such that $h_{d,r}$ is not birational. We prove this conjecture for $r=1$ (the case of binary forms). We study the restriction of the Hessian map to the locus of hypersurfaces of degree $d$ with Waring rank $r+2$ in $\mathbb{P}^r$, proving that this restriction is injective as soon as $r\geq 2$, which implies that $h_{3,3}$ is birational onto its image. We prove that the Hessian map is generically finite, unless the case $h_{3,1}$ .

Laura Pertusi (Università degli Studi di Milano): Serre-invariant stability conditions and cubic threefolds

Abstract: Stability conditions on the Kuznetsov component of a Fano threefold of Picard rank 1, index 1 and 2 have been constructed by Bayer, Lahoz, Macrì and Stellari, making possible to study moduli spaces of stable objects and their geometric properties. In this talk we investigate the action of the Serre functor on these stability conditions. In the index 2 case and in the case of GM threefolds, we show that they are Serre-invariant. Then we prove a general criterion which ensures the existence of a unique Serre-invariant stability condition and applies to some of these Fano threefolds. Finally, we apply these results to the study of moduli spaces in the case of a cubic threefold X. In particular, we prove the smoothness of moduli spaces of stable objects in the Kuznetsov component of X and the irreducibility of the moduli space of stable Ulrich bundles on X. These results come from joint works with Song Yang, with Soheyla Feyzbakhsh and with Ethan Robinett.

Andrea Ricolfi (Università di Bologna): A tale of two d-critical structures

Abstract: D-critical schemes and Artin stacks were introduced by Joyce in 2015, and play a central role in Donaldson-Thomas theory. They typically occur as truncations of (-1)-shifted symplectic derived schemes, but the problem of constructing the d-critical structure on a "DT moduli space" without passing through derived geometry (which is hard) is wide open. We discuss this problem, and new results in this direction, when the moduli space is the Hilbert (or Quot) scheme of points on a Calabi-Yau 3-fold. Joint work with Michail Savvas.

Benjamin Schmidt (Leibniz Universität Hannover): Sheaves of low rank in three dimensional projective space

Abstract: Moduli spaces of vector bundles, or more generally sheaves on algebraic varieties, are usually badly behaved. As soon as the dimension of the variety is at least three, they satisfy Murphy's Law in algebraic geometry, i.e., all types of singularities can occur on them. In this talk, I will introduce a class of sheaves in three-dimensional projective space and conjecture that their moduli spaces are smooth and irreducible, contrary to the general picture. This conjecture can be proved for sheaves of low rank. Everything is very closely related to the numerical study of Chern characters of semistable sheaves.

Alexander Tikhomirov (HSE University): New moduli component of rank 2 bundles on projective space

Abstract: We present a new family of monads whose cohomology is a stable rank two vector bundle on $\mathbb{P}^3$. We also study the irreducibility and smoothness together with a geometrical description of some of these families. These facts are used to construct a new infinite series of rational moduli components of stable rank two vector bundles with trivial determinant and growing second Chern class. We also prove that the moduli space of stable rank two vector bundles with trivial determinant and second Chern class equal to 5 has exactly three irreducible rational components. This is a joint work with Ch. Almeida, M. Jardim and S. Tikhomirov.