Next Seminar: July 13 14:15

Patricio Gallardo (Riverside)

On Moduli of Quiver Representations and Applications to Geometry

Title and Abstract Link Teams
Abstract: I will present an equidistribution result for families of (non-degenerate) subvarieties in a family of abelian varieties. Using this result, one can deduce a uniform version of the classical Bogomolov conjecture for curves embedded in their Jacobians, namely that the number of torsion points lying on them is uniformly bounded in the genus of the curve. This has been previously only known in a few select cases by work of David-Philippon and DeMarco-Krieger-Ye. Furthermore, one can deduce a rather uniform version of the Mordell conjecture by complementing a result of Dimitrov-Gao-Habegger: The number of rational points on a smooth algebraic curve defined over a number field can be bounded solely in terms of its genus and the Mordell-Weil rank of its Jacobian. Again, this was previously known only under additional assumptions (Stoll, Katz-Rabinoff-Zureick-Brown). All these results have been recently generalized beyond curves in joint work with Ziyang Gao and Tangli Ge, but I will focus on the simpler case of curves.
Abstract: Smooth minimal surfaces of general type with \(K^2=1,\ p_g=2\), and \(q=0\) constitute a fundamental example in the geography of algebraic surfaces, and the 28-dimensional moduli space \(\mathbf{M}\) of their canonical models admits a modular compactification \(\overline{\mathbf{M}}\) via the minimal model program. We describe eight new irreducible boundary divisors in such compactification parametrizing reducible stable surfaces. Additionally, we study the relation with the GIT compactification of \(\mathbf{M}\) and the Hodge theory of the degenerate surfaces that the eight divisors parametrize. This is joint work with Patricio Gallardo, Gregory Pearlstein, and Zheng Zhang.
Abstract: Inspired from ideas in topology, Koszul modules turned out to have important algebro-geometric applications for instance to (i) Green's Conjecture on syzygies of canonical curves, (ii) stabilization of cohomology of projective varieties in arbitrary characteristics and (iii) a resolution of an effective form of an important conjecture of Suciu's on Chen invariants of hyperplane arrangements. I will discuss new developments related to this circle of ideas obtained in joint work with Aprodu, Raicu and Suciu.
Abstract: One says a scheme, or an algebraic stack, has the resolution property if every coherent sheaf is the quotient of a locally free sheaf. Although this is a fundamental and widely used property in algebraic geometry, it is still poorly understood. After giving the appropriate definitions, we will explain the two most important sources of non-examples: (1) affine group schemes G/S which cannot be embedded into GL_n but which are forms of embeddable group schemes, and (2) cohomological Brauer classes which are not represented by Azumaya algebras. After describing a new way to construct non-trivial vector bundles on schemes and stacks, we introduce the notion of an R-unipotent morphism and characterize it geometrically. We will then present a surprising local to global principle: a locally R-unipotent morphism over a base with enough line bundles is globally R-unipotent. To conclude, we will explain why the unipotent analogues of (1) and (2) above cannot occur.
Abstract: I will report on some recent results on the geometry of irreducible symplectic varieties that are deformation of moduli spaces of sheaves on K3 surfaces. In the first part of the talk I will introduce the main characters of this story and I will talk about their deformations: this will also include a joint work in progress with A. Perego and A. Rapagnetta on (undesingularisable) singular moduli spaces of sheaves. In the second part I will mostly focus on moduli spaces of O'Grady type and their symplectic desingularisation, presenting an older result on their ample cone (joint with G. Mongardi), and illustrating a recent application to symplectic automorphisms (joint with L. Giovenzana, A. Grossi and D.C. Veniani).
Abstract: I will discuss joint work with Chris Peters which extends rigidity results of Arakalov, Faltings and Peters to period maps arising from families of complex algebraic varieties which are non-necessarily proper or smooth. Inspired by recent work with P. Gallardo, L. Schaffler, Z. Zhang, I will discuss two classes of elliptic surfaces which can be presented as hypersurfaces in weighted projective spaces which have a unique canonical curve. In each case, we will show that infinitesimal Torelli fails for \(H^2\) of the compact surface, but is restored when one considers the period map for the complement of the canonical curve. We will also show that these period maps are rigid, in the sense that they do not admit any horizontal deformations which keep the source and target fixed.
Abstract: We know by Falting's theorem that a curve \(C\) of genus \(g>1\) defined over the rationals has a finite number of rational points, but there is no general procedure to provably compute the set \(C(\mathbb{Q})\). When the rank of the Mordell-Weil group \(J(\mathbb{Q})\) (with \(J\) the Jacobian of \(C\)) is smaller than \(g\) we can use Chabauty method, i.e. we can embed \(C\) in \(J\) and, after choosing a prime p, we can view \(C(\mathbb{Q})\) as a subset of the intersection of \(C(\mathbb{Q}_p)\) and the closure of \(J(\mathbb{Q})\) inside the p-adic manifold \(J(\mathbb{Q}_p)\); this intersection is always finite and computable up to finite precision. Minhyong Kim has generalized this method by inspecting (possibly non-abelian) quotients of the fundamental group of \(C\). His ideas have been made effective in some new cases by Balakrishnan, Dogra, Muller, Tuitman and Vonk: their "quadratic Chabauty method" works when the rank of the Mordell-Weil group is strictly less than \(g + s -1\) (with s the rank of the Neron-Severi group of \(J\)). In the seminar we will give a reinterpretation of the quadratic Chabauty method, only using the Poincaré torsor of \(J\) and a little of formal geometry, and we will show how to make it effective. This is joint work with Bas Edixhoven.
Abstract: A Hermitian manifold is locally conformally Kaehler (LCK) if it admits a Kaehler cover on which the deck group acts by homotheties. If this Kaehler metric has a positive, global potential, the manifold is called LCK with potential. The typical example is the Hopf manifold which is clearly non-algebraic. However, we prove that the coverings of LCK manifolds with potential have an algebraic structure, being in fact affine cones over projective orbifolds. This permits using algebraic geometry techniques in the study of non-algebraic manifolds. The material that I shall present belongs to joint works with Misha Verbitsky.
Abstract: Since the seminal papers of Xiao and Cornalba-Harris on fibred surfaces in the '80's, slope inequalities have been a central problem in algebraic geometry, leading both to geographical results on the invariants of varieties, and to results about the positive cones of divisors of certain moduli spaces. I will describe the problem in general, the Cornalba-Harris approach (and its limits), and some recent results obtained in collaboration with Miguel Angel Barja, regarding the slope of fibrations whose general fibres are complete intersections. In particular, we prove the full slope ineqaulity in any dimension for families whose general fibre is a local complete intersection of stable (e.g. smooth) hypersurfaces. Eventually, if time permits, I will describe also a way of finding, in the particular case of a family which is a global complete intersection, an instability result.
The conference "WINTER MEETING IN ALGEBRA AND GEOMETRY 2022" is a Winter meeting in Algebra and Geometry in Rome that takes place in the rione Monti. For more info here is the Link to the Official Webpage
Abstract: The moduli space \(\mathcal{M}_g\) of curves of genus \(g\) is a central object in algebraic geometry and our main goal is to study it from the point of view of one of the most important topological invariant, its rational cohomology. For \(g\geq5\), the rational cohomology of \(\mathcal{M}_g\) is still unknown and one way to approach this problem is by computing first the rational cohomology of some locus inside \(\mathcal{M}_g\), such as the trigonal one \(\mathcal{T}_g\). In the first part of the talk, we will study the rational cohomology of \(\mathcal{T}_g\). Specifically, we prove that, similarly to the cohomology of \(\mathcal{M}_g\), the cohomology ring of \(\mathcal{T}_g\) stabilizes and it coincides with the tautological ring within the stable range. This will be done by studying the natural embedding of trigonal curves in Hirzebruch surfaces and by using Gorinov-Vassiliev's method for the cohomology of complements of discriminants. In the second part, we will inspect what happens for curves, which are also embedded in Hirzebruch surfaces but have gonality different from 3. In particular I will discuss a joint work with Jonas Bergström which describes the stable cohomology of the moduli space of gonality 2 curves, embedded on a Hirzebruch surface of fixed degree, and how this moduli space relates to the one parametrizing hyperelliptic curves with marked points.
Abstract: We study the slope of modular forms on the Siegel space. We will recover known divisors of minimal slope for \(g\leq 5\) and we discuss the Kodaira dimension of the moduli space of principally polarized abelian varieties \(\mathcal{A}_g\) (and eventually of the generalized Kuga's varieties). Moreover we illustrate the cone of moving divisors on \(\mathcal{A}_g\). Partly motivated by the generalized Rankin-Cohen bracket, we construct a non-linear holomorphic differential operator that sends Siegel modular forms to Siegel cusp forms, and we apply it to produce new modular forms. Our construction recovers the known divisors of minimal moving slope on \(\mathcal{A}_g\) for small genera.
Abstract: I will illustrate a description of torsion points on a theta divisor (of a complex principally polarized abelian variety) making use of certain semihomogeneous vector bundles introduced and studied by Mukai and D. Oprea. As a consequence, I will show an upper bound on the number of n-torsion points on a theta divisor (for a fixed positive integer n). The bound is achieved if and only if the p.p.a.v. is a product of elliptic curves, proving a conjecture of Auffarth, Marcucci, Pirola and Salvati Manni. Partly a joint work with Riccardo Salvati Manni.
Abstract: In this talk I will explain how to recognize complex tori among Kähler klt spaces (smooth in codimension 2) in terms of vanishing of Chern numbers. It requires first to define Chern classes on singular spaces (a rather unstable notion). On the way, we will establish a singular version of the Bogomolov-Gieseker inequality for stable sheaves and study what can be said in the equality case. Joint work with Patrick Graf and Henri Guenancia.
Abstract: The moduli space of curves (by which I mean its Deligne-Mumford compactification) is a well studied object in algebraic geometry. Mumford introduced the notion of tautological intersection theory to study a part of the intersection theory which is simple enough to be tractable, but rich enough to be meaningful. Hodge integrals are a class of tautological intersection numbers that arise from intersecting the chern classes of the Hodge bundle and of the cotangent line bundles. In the first part of the talk I will introduce all these concepts and review some "classical" structural results about Hodge integrals. When running the MMP on the moduli space of curves, after the first wall-crossing one sees the moduli space of pseudo stable curves, which is the target of a birational regular morphism from the moduli space of curves. We investigate how the Hodge bundles on either side of this morphism are related, and how, correspondingly, there are very rich combinatorial relation between Hodge integrals and pseudo stable Hodge integrals. This talk is based on joint work with Gallegos, Ross, Wise, Van Over and on some of Matthew Williams' doctoral work.
Abstract: We show that the cohomology of moduli spaces of Higgs bundles decomposes in elementary summands depending on the topology of the symplectic singularities on a (fixed!) master object and/or the combinatorics of certain posets and lattice polytopes. This is based on a joint work with Luca Migliorini and Roberto Pagaria.
Abstract: La congettura SYZ venne formulata negli anni novanta da Strominger, Yau e Zaslow come spiegazione geometrica dei fenomeni di mirror symmetry. Kontsevich e Soibelman hanno proposto un approccio non-archimedeo alla congettura, che un recente lavoro di Li ha messo in stretta relazione con la congettura originaria.  In questo seminario mi concentrerò sul caso delle ipersuperfici di Calabi-Yau in P^n. In collaborazione con Jakob Hultgren, Mattias Jonsson e Nick McCleerey, risolviamo una congettura non-archimedea proposta da Li e deduciamo che le fibrazioni SYZ esistono su un aperto arbitrariamente grande dell'ipersuperficie.
Abstract: The representation varieties of a Dynkin quiver are examples of affine varieties (actually affine spaces) over which a structure group acts with finitely many orbits, by Gabriel's theorem. To describe orbit closures there are invariants which generalize the usual ranks. After recalling this classical situation, I will consider the case when the quiver is endowed with a self-duality and hence the representation varieties are endowed with an involution: in this case it is interesting to consider the subvariety of fixed points for this involution. It is acted upon by a group that it is the subgroup of fixed points for an involution of the structure group. How can one describe the orbit closures of such fixed points? It turns out that the orbit closures are nothing else than the intersection of the orbit closures for the "big" group with the variety of fixed points. This is the main result that I want to present which I obtained in collaboration with Magdalena Boos (arXiv 2106.08666).
Abstract: Ulrich bundles are certain arithmetically Cohen Macaulay bundles on projective varieties enjoying special cohomological properties. Their study originates in a paper by B. Ulrich in 1984 in the framework of commutative algebra and the attention of algebraic geometers was drawn by a landmark paper by Eisenbud-Schreyer-Weyman, where, among other things, the Chow form of a projective variety X is computed using Ulrich bundles on X, if they exist. It is furthermore conjectured that any projective variety should carry an Ulrich bundle. This conjecture remains widely open in general, as well as related questions regarding possible ranks of Ulrich bundles, their stability and moduli, although a lot is known for specific classes of varieties (e.g., curves, Segre and Grassmann varieties, rational scrolls, complete intersections, some classes of surfaces and threefolds, etc.) In the talk I will present recent results concerning existence, ranks, stability and moduli of Ulrich bundles on Fano threefolds, obtained in collaboration with C. Ciliberto and F. Flamini (arXiv:2206.09986 and arXiv:2205.13193), extending and unifying results by (many) other authors.
Abstract: I will start by reviewing how to compactify the universal Jacobian stack, parametrizing pointed curves endowed with a line bundle, over the moduli stack of stable pointed curves. I will then introduce two incarnations of tropical universal Jacobian and discuss their relationships with the compactified universal Jacobian: one via non-archimedean geometry and one via logarithmic geometry. This is a joint work (partly in progress) with M. Melo, S. Molcho, M. Ulirsch, J. Wise.
Abstract: We will explain a construction of Albanese maps for orbifolds (or C-pairs) with applications to hyperbolicity such as a generalization of the Bloch-Ochiai theorem. (Joint with Stefan Kebekus).
Abstract: Enriques manifolds are non simply connected manifolds whose universal cover is irreducible holomorphic symplectic, and as such they are natural generalizations of Enriques surfaces. In this talk I will report on a joint work with Alessandra Sarti in which we study the Morrison-Kawamata cone conjecture for such manifolds using the analogous result (established by Amerik-Verbitsky) for their universal cover.
The "The SEVENTH MINI SYMPOSIUM of the Roman Number Theory Association" is the Seventh instance of a meeting in Number Theory that this year takes place in the rione Monti in Rome. This year it will include an atelier LEAN. For more info here is the Link to the Official Webpage
Abstract: Level structures are extra data that can be added to some moduli problems in order to rigidify the situation. For example, in the case of curves, they yield smooth Galois covers of the moduli space M_g, and the problem of extending this picture to the boundary was studied by several authors, using in particular admissible covers and twisted curves. I will report on some work in progress with M. Ulirsch and D. Zakharov, in which we consider a tropical notion of level structure on a tropical curve. The moduli space of these is expected to be closely related to the boundary complex of the stack of G-admissible covers. As usual, logarithmic geometry stands in the middle and provides a convenient language to bridge the two worlds.
Abstract: "Hyperelliptic Odd Coverings" are a class of odd coverings of \(C \to \mathbb{P}^1\), where C is a hyperelliptic curve. They are characterized by the behavior of the hyperelliptic involution of C with respect to an involution of \(\mathbb{P}^1\). I will talk about some ways for studying this type of coverings: by fixing an effective theta characteristic, they correspond to the solutions of a certain type of differential equations. Considering them as elements in a suitable Hurwitz space, they can be characterized using monodromy and then studied from the point of view of deformations. When C is general in H_g, the number of possible Hyperelliptic Odd Coverings \(C \to \mathbb{P}^1\) of minimum degree is finite. The main result will be how to compute this number. This is a work in collaboration with Gian Pietro Pirola. In the first part of the talk, I will introduce the Hyperelliptic Odd Coverings from a geometric point of view, contextualizing them in the panorama of other enumerative works (in collaboration with Farkas, Naranjo, Pirola, Lian). In the second part of the talk I will talk about some proofs and some open problems.
Abstract: The existence of Kähler-Einstein metrics on Fano 3-folds can be determined by studying some positive numbers called stability thresholds. K-stability is ensured if appropriate bounds can be found for these thresholds. An effective way to verify such bounds is to construct flags of point-curve-surface inside the Fano 3-folds. This approach was initiated by Abban-Zhuang, and allows us to restrict the computation of bounds for stability thresholds only on flags. We employ this machinery to prove K-stability of terminal quasi-smooth Fano 3-fold hypersurfaces. Many of these varieties had been attacked by Kim-Okada-Won using log canonical thresholds. In this talk I will tackle the remaining Fano hypersurfaces via Abban-Zhuang Theory.
Abstract: I will give a brief introduction on weighted blow-ups and on the relevant results from intersection theory for ordinary blow-ups. When looking at the generalization of weighted blow-ups, some caution is needed as we don't have access to all of the tools used in the classical case. One needs, for example, to take into account that the weights make for some non reducedness or some complications with torsion in the Chow rings. I will give a general presentation for the Chow ring of the exceptional divisor of a weighted blow-up and for the Gysin homomorphism induced by it. These will allow to give a presentation for the integral Chow ring of a weighted blow-up. If time permits, I will use this presentation to compute the Chow ring of \(\overline M_{1,2}\).
Abstract: The notion of residual intersections was introduced by Artin and Nagata. Roughly speaking, given an algebraic variety X and a closed subscheme Y in X, which is contained in another closed subscheme Z, then a closed subscheme W such that \(W \cup Y = Z\) is a residual intersection of Y in Z. This idea can be formalized as follows: Let I be an ideal in a local Cohen-Macaulay ring R, and \(A = (a_1, \ldots, a_s) \subsetneq I\). Then \(J = A:I\) is called an s-residual intersection of I if \(ht(J) \geq s \geq ht(I)\). Residual intersections provide a generalization of linkage. Indeed, if \(J = A:I\) and \(I = A:J\) for A a regular sequence, I and J are said to be linked. I will show how results of Huneke and of Kustin and Ulrich on residual intersections for standard deteminantal ideals and Pfaffian ideals respectively arise in the context of ideals of Schubert varieties in the big opposite cell of homogeneous spaces. This is joint work with J. Torres and J. Weyman.
Abstract: Let \(X\) be a compact, complex surface of general type whose cotangent bundle \(\Omega_X\) is strongly semi-ample. We study the pluri-cotangent maps of \(X\), namely the morphisms \(\psi_n \colon \mathbb{P}(\Omega_X) \to \mathbb{P}(H^0(X, \, S^n \Omega_X))\) defined by the vector space of global sections \(H^0(X, \, S^n \Omega_X)\). This is joint work with X. Roulleau.

The seminar is usually held every Thursday at 14:15-15:45 in Aula M1. This year seminars will be held in a blended format: in person (for more info you can send an email to an organizer) and on the platform Microsoft Teams (online).

The seminar is organized by Luca Schaffler and Amos Turchet and maintained by the Geometry Group of the Department of Mathematics and Physics at the Roma Tre University.

We acknowledge the support of the grants PRIN2020: Curves, Ricci flat varieties and their interactions, PRIN2017: Advances in Moduli Theory and Birational Classification, and the support of the Department of Mathematics and Physics at Roma Tre University.

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