Seminars 2020 - 2021

**Abstract**: Un fibrato di Ulrich su una varietà \(X\) immersa in \(\mathbb{P}^N\) è un fibrato vettoriale \(E\) tale che \(H^i(E(-p))=0\) per \(i \ge 0\) e \(1 \le p \le dim X\). Un po' come la regolarità di Castelnuovo-Mumford, questa apparentemente semplice definizione (Mumford dixit) comporta invece parecchie conseguenze interessanti per la geometria di \(X\). Ne ricorderemo alcune e passeremo poi a descrivere due aspetti: a) l'esistenza dei fibrati di Ulrich, nota solo per le curve, ma congetturata in generale; b) la positività dei fibrati di Ulrich e le sue conseguenze.

**Abstract**: An important open problem in arithmetic geometry is to identify geometric properties of a variety \(X\) defined over \( \mathbb{Q}\) that guarantee that the set \(X(k)\) is dense for some finite extension \(k\) of \(Q\) (or that admit dense entire curves). We will explain how the usual geometric classification in dimension at least 2 does not provide an adequate answer and present a new type of classification proposed by Campana that tries to answer these type of questions. In particular we will introduce the class of “special varieties” in terms of Bogomolov sheaves, and in terms of Campana Orbifolds, and explain the canonical (and functorial) decomposition of varieties in their “special” and “non-special” parts. Finally we will show how this questions are related to questions of Harris and Abramovich-Colliot-Thélène and discuss interesting constructions in dimension 3. (Based on joint work with Erwan Rousseau and Jiulie TY Wang)

**Abstract**: We will discuss various questions about existence of curves with negative self-intersections on complex projective surfaces. In particular, we will present a recent result that shows the existence of smooth complex projective surfaces containing reduced, irreducible curves \(C\) of arbitrarily negative self-intersection \(C^2\). Previously the only known examples of surfaces for which \(C^2\) was not bounded below were in positive characteristic, and the Bounded Negativity Conjecture, dating back to Federigo Enriques, predicted that no examples could arise over the complex numbers. Our examples are special types of Hilbert modular surfaces. This is joint work with Th. Bauer, B. Harbourne, A. Kuronya, S. Muller-Stach, and T. Szemberg. .

**Abstract**: We investigate a refined Derived Torelli Theorem for Enriques surfaces. Namely, we prove that two (generic) Enriques surfaces are isomorphic if and only if their Kuznetsov components are Fourier-Mukai equivalent. We analyze the similarities with analogous results for cubic fourfolds and threefolds and we show the applications of our techniques to a conjecture by Ingalls and Kuznetsov about the derived categories of Artin-Mumford quartic double solids. This part is a joint work with Li, Nuer and Zhao. If time permits, I will explain some on-going joint work with Li and Zhao where we remove the genericity assumption in our refined Derived Torelli Theorem.

**Abstract**: La quartica di Igusa, o quartica di Castelnuovo-Richmond, è uno speciale quartic threefold, noto per la la sua ubiquità in geometria algebrica. Nel seminario verrà descritto un nuovo esempio di tale ubiquità e cioè la relazione tra quartica di Igusa e mappa di Prym in genere 6. Tale mappa \(P: \mathcal{R}_6 \to \mathcal{A}_5\), descritta da Donagi e Smith, domina lo spazio dei moduli \(\mathcal{A}_5\) delle varietà abeliane p.p. di dimensione 5. \(P\) ha grado 27 e la sua monodromia identifica la fibra generica di \(P\) con la configurazione delle 27 rette di una superficie cubica. La stessa monodromia permette di costruire, a partire da \(P\), altre mappe associate a \(P\) sulle quali poco è noto. Tra queste ha particolare rilievo la mappa \(J: D \to \mathcal{A}_5\), che ha come fibra la configurazione dei 36 insiemi di 'double-six' di rette della superficie cubica. Nel seminario verrà descritta la mappa \(J\) e la geometria della sua relazione con la quartica di Igusa \(B\): si proverà che \(D\) è birazionale con lo spazio dei moduli dei quartic threefolds, (con 30-nodi), che tagliano su \(B\) una sezione quadratica doppia e che \(J\) è la naturale mappa dei periodi associata. Si proverà inoltre la razionalità di \(D\).

**Abstract**: A spherical metric on a surface is a metric of constant curvature 1, which thus makes the surface locally isometric to \(S^2\). Such a metric has a conical point \(x\) of angle \(2\pi\theta\) if its area element vanishes of order \(2(\theta-1)\) at \(x\). If the conformal class is prescribed, a spherical metric can be viewed as a solution of a suitable singular Liouville equation. If the conformal class is not prescribed, isotopy classes of spherical metrics can be considered as flat \((SO(3,R),S^2)\)-structure, and so their deformation space has a natural finite-dimensional real-analytic structure. Additionally, the moduli space of spherical surfaces of genus \(g\) with \(n\) conical points comes endowed with a natural forgetful map to the moduli space of Riemann surfaces of genus \(g\) with \(n\) marked points. I will begin by giving an overview of what is known about the topology of the moduli space of spherical surfaces and the above mentioned forgetful map. I will then focus on the case of genus 1 with 1 conical point (joint works with Eremenko-Panov and with Eremenko-Gabrielov-Panov).

**Abstract**: I will discuss a recent project in computing the top weight cohomology of the moduli space \(\mathcal{A}_g\) of principally polarized abelian varieties of dimension \(g\) for small values of \(g\). This piece of the cohomology is controlled by the combinatorics of the boundary strata of a compactification of \(\mathcal{A}_g\). Thus, it can be computed combinatorially. This is joint work with Juliette Bruce, Melody Chan, Margarida Melo, Gwyneth Moreland, and Corey Wolfe.

**Abstract**: Given a triangulated category \( D\), one can consider its space of stability conditions and a theory of Donaldson-Thomas (DT) invariants counting semistable objects. One of the conjectures of Mirror Symmetry leads to looking for geometric structures parametrized by the stability space, and a way of investigating these structures is by stating and solving a Riemann-Hilbert-Birkhoff boundary problem induced by the wall-crossing formula for DT data. In the case of a very simple DT theory, the solution to this problem is given in terms of Barnes multiple Gamma functions. Based on a joint work with T.Bridgeland and J.Stoppa.

**Abstract**: Let \(S\) be a smooth algebraic surface in \(\mathbb{P}^3\). A curve \(C\) in \(S\) has a cohomology class \([C]\) in \(H^1( \Omega^1_S)\). The Hodge class of \(C\) is the equivalence class of \([C]\) in the quotient modulo the subspace generated by the class of a plane section of \(S\). Recently, Movasati and Sertoz pose several interesting questions about the reconstruction of \(C\) from its Hodge class. In particular they give the notion of a perfect class: the Hodge class of a curve \(C\) is perfect if its annihilator is a sum of ideals of curves \(C\) whose Hodge class is a nonzero rational multiple of that of \(C\). I will report on a joint work with Gian Pietro Pirola and Enrico Schlensiger, in which we give an answer to some of these questions: we show that the Hodge class of a smooth rational quartic on a surface of degree 4 is not perfect, and that the Hodge class of an arithmetically Cohen-Macaulay curve is always perfect.

**Abstract**: Consider the non-trivial part of a semi-orthogonal decomposition of the derived category of a cubic threefold, called Kuznetsov component. In this talk, I will first examine S-invariant Bridgeland stability conditions on the Kuznetsov component which are basically unchanged after the action of the Serre functor \(S\). Then I will describe certain moduli spaces of stable objects with respect to these stability conditions. As an application, I will show a categorical version of the Torelli theorem for cubic threefolds, which says that the cubic threefold can be recovered from its Kuznetsov component. This talk is based on joint work in progress with Laura Pertusi and a group project with A.Bayer, S.V. Beentjes, G. Hein, D. Martinelli, F. Rezaee and B. Schmidt.

**Abstract**: Character varieties parametrise representations of the fundamental group of a compact Riemann surface, and in general they are singular affine varieties. They are prototypes of singular hyperkahler varieties and provide examples of SYZ mirror pairs. In this seminar, we will present the computation of the intersection cohomology of rank two character varieties. We will derive several results concerning the P=W conjecture and the Hausel-Thaddeus topological mirror symmetry conjecture for singular character varieties.

**Abstract**: We construct an explicit complex smooth Fano threefold with Picard numer 1, index 1 and degree 10 (a Gushel-Mukai threefold) and prove that it is not rational by showing that its intermediate jacobian has a faithful \(\mathrm{PSL}(2,F_{11})\) action. The construction is based on a very special double EPW sextic. This is joint work with O. Debarre..

**Abstract**: Negativity of the holomorphic sectional curvature is a sufficient condition which insures that a compact complex manifold is Kobayashi hyperbolic, but it is far from being necessary. After reviseing the essential notions and concepts of curvature for Hermitian and/or Kähler manifolds, and how they are linked with Kobayashi hyperbolicity, we shall quantify the above-mentioned lack of necessity and illustrate it also via examples. In the second part of the talk, we shall explain how it is possible, under the stronger hypothesis of negativity of the holomorphic sectional curvature, to prove a longstanding conjecture by Kobayashi stating that a compact Kobayashi hyperbolic Kähler manifold is projective and canonically polarized (works of Wu-Yau, Tosatti-Yang, Diverio-Trapani, and Guenancia).

**Abstract**: In characteristic zero birational algebraic geometry, Kawamata-Viehweg vanishing is a centrally important tool. For some applications in characteristic \(p > 0\), one may use Frobenius and perturbations as a replacement for resolution of singularities and Kawamata-Viehweg vanishing. This talk will show how to use Bhatt's vanishing theorem for absolute integral closures mixed characteristic as a replacement for resolutions and Kawamata-Viehweg vanishing theorems in a number of applications. This is joint work with B. Bhatt, L. Ma, Z. Patakfalvi, K. Tucker, J. Waldron and J. Witaszek.

**Abstract**: Dopo una doverosa introduzione, la prima parte del seminario sarà dedicata alla teoria dell'intersezione dei divisori di cubiche speciali nello spazio dei moduli delle ipersuperfici cubiche di dimensione 4 (aka cubic fourfolds). Daremo delle condizioni necessarie affinché (fino a) 20 divisori si intersechino, e descriveremo le superfici \(K3\) associate a queste cubiche - nel senso della teoria di Hodge. Applicheremo questa costruzione per costruire nuove famiglie di cubiche con Chow motive di dimensione finita e di tipo abeliano. Infine, considereremo alcune varietà di HyperKähler associate alle cubiche (la varietà di Fano delle rette contenute nel 4fold, il LLSvS 8fold, ecc.) e mostreremo che in alcuni casi i nostri precedenti risultati implicano che anche queste varietà di HK hanno un Chow motive finito dimensionale.

**Abstract**: In this talk I will discuss geometric properties of certain vector bundles on the moduli space of stable \(n\)-pointed curves which arise from modules over vertex operator algebras. I will in particular describe conditions on the vertex algebra that guarantee that the factorization property holds for these vector bundles and discuss its consequences and open problems. This is based on a joint work with A. Gibney and N. Tarasca.

**Abstract**: This is joint work with with János Kollár, Martin Olsson, and Will Sawin. I will discuss an extension of the classical Veblen-Young theorem on axiomatic projective geometry to arbitrary algebraic varieties. In particular, I will explain how most proper, normal varieties are uniquely determined (up to isomorphisms of the base field) from their underlying Zariski topological spaces.

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13 May
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13 May

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**Abstract**: The degree \(d\) universal Jacobian parametrizes degree \(d\) line bundles on smooth curves. There are several approaches on how to extend it to a proper family over the moduli space of stable curves. In this talk, we introduce a simple definition of a fine compactified universal Jacobian. We focus on the case of genus 1 and obtain a combinatorial classification for fine universal compactified Jacobians, which enables us to construct new examples of them. The description we obtain for universal fine compactified Jacobians of genus 1 also yields a formula for their rational cohomology. This is joint work with Nicola Pagani (Liverpool).

**Abstract**: The problem of determining the birational nature of the moduli space of curves of genus \(g\) has received constant attention in the last century and inspired a lot of development in moduli theory. I will discuss progress achieved in the last 12 months. In particular, making essential of tropical methods it has been showed that both moduli spaces of curves of genus 22 and 23 are of general type (joint with D. Jensen and S. Payne).

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3 Jun
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3 Jun

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**Abstract**: Showing that a given exceptional collection in a triangulated category is full is generally a nontrivial problem. In the numerically finite case, an obvious necessary condition is that the exceptional collection is numerically full, i.e., it spans the numerical Grothendieck group of the triangulated category. This condition is however not sufficient, due to the possible presence of phantom subcategories, i.e., nontrivial subcategories that are invisible to numerical detection. A Bridgeland stability condition on the right orthogonal of the exceptional collection (when it exists), including a “positivity condition” for nonzero objects, notably forbids the presence of these phantoms thus implying fullness. As an explicit application I'll show how one can use this to rediscover Kapranov's full exceptional collection on a smooth quadric threefold. The same technique can be used to recover a few classical results on the commutativity of certain Kuznetsov components, e.g., Bondal-Orlov's proof of the commutativity of \(Ku(Y_4)\) and the Addington-Thomas theorem on cubic fourfolds. Joint work with Barbara Bolognese (arXiv:2103.15205)

**Abstract**: The Betti numbers of Hilbert schemes of points in the plane were first computed by Ellingsrud and Strømme. In this talk, we'll let a finite cyclic group act on the plane, and consider the analogous problem of computing the Betti numbers of the fixed loci under the induced action on the Hilbert scheme of points. The main result is a new product formula for the Betti numbers in the case where no element of the group fixes the symplectic form on the plane. This is joint work with Paul Johnson.

Info

The seminar is usually held every **Thursday at 14:30-16:00**. Due to the pandemic the next seminars will be held online on the platform Microsoft Teams.

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