Seminars 2021 - 2022

**Abstract**: Bruhat-Tits buildings are natural piecewise-linear analogues of symmetric spaces for semi-simple (or more generally reductive) groups. In this talk we show how a variation on the theme of Bruhat-Tits buildings can be used to define a natural tropicalization of a reductive group \( G\). We explain how the choice of a stacky fan structure on the tropicalization gives rise to a toroidal bordification of \( G\) and how to prove a version of faithful tropicalization. Finally, we combine this story with methods from logarithmic geometry to give a new perspective on parabolic principal bundles.

**Abstract**: The classical Abel-Jacobi theorem gives necessary and sufficient conditions for a divisor on a Riemann surface to be the divisor of zeros and poles of a rational function. The modern counterpart of this question leads to the study of the "double ramification cycles", which are, roughly speaking, the loci of curves in \(\overline{M}_{g,n}\) which admit a rational function with prescribed zeros and poles along their markings. The DR cycles have been extensively studied in recent years for a variety of reasons -- their connection with Gromov-Witten theory, strata of meromorphic differentials. In this talk I will discuss how approaching the problem from the perspective of logarithmic geometry reveals additional structure on these cycles, and how it naturally connects them to the theory of universal stability conditions, yielding new formulas in the process. This is joint work with Holmes, Pandharipande, Pixton and Schmitt.

**Abstract**: Let \( (S,L)\) be a general K3 surface of genus \(g\). I will prove that the closure in \(|L|\) of the Severi variety parametrizing curves in \(|L|\) of geometric genus \(h\) is connected for \(h\geq 1\) and irreducible for \(h\geq 4\), as predicted by a well known conjecture. This is joint work with Andrea Bruno.

**Abstract**: Given integers \(g \geq 0, n \geq 1\), and a vector \( w \in (\mathbb{Q} \cap (0, 1] )^n\) such that \( 2g -2 + \sum w_i > 0 \), we study the topology of the moduli space \(\Delta_{g,w}\) of \(w\)-stable tropical curves of genus \(g\) with volume 1. The space \( \Delta_{g,w} \) is the dual complex of the divisor of singular curves in Hassett's moduli space of \(w\)-stable genus \(g\) curves \(\mathcal{M}_{g,w} \). When \( g\geq 1 \), we show that \( \Delta_{g,w} \) is simply connected for all possible \(w\). We also give a formula for the Euler characteristic of \(\Delta_{g,w}\) in terms of the combinatorics of \(w\).

**Abstract**: Bridgeland stability conditions have proved to be an extremely versatile tool to study the birational geometry of classical moduli spaces of sheaves, to solve counting problems, to help define new structure in mirror symmetry. After a gentle introduction, we will focus on the relation between stability conditions and exceptional collections. A powerful tool of investigation of Fano varieties is provided by exceptional collections in their derived categories. In general, proving the fullness of such a collection is a hard problem, often done on a case-by-case basis, with the aid of a deep understanding of the underlying geometry. Likewise, when an exceptional collection is not full, it is not straightforward to determine whether its residual category is the derived category of a variety. Taking after Bondal and Orlov, we examine two cases: the case of quadric hypersurfaces in \( \mathbb{P}^{n+1}\) and the case of the index 2 Fano threefold \(Y\) (the generic intersection of two quadrics in \(\mathbb{P}^5\). In the first case, we prove that the classical result by Kapranov on the fullness of the standard exceptions is equivalent to the existence of a numerical stability condition on the residual category of the exceptional collection of the quadric. In the second case, we show how the same technique recovers the equivalence of the residual category of the exceptional collection \(\{\mathcal{O}_Y, \mathcal{O}_Y (1)\}\) with the derived category of a genus 2 curve. This is joint work with Domenico Fiorenza. Moreover, we will introduce ideas on methods to construct stability conditions via exceptional collections and hearts of finite length from a joint work in progress with Domenico Fiorenza and Alex Küronya.

**Abstract**: We present briefly the theory of divisors on graphs, parallel to that of line bundles on curves. We show some tools for computing the ranks of spin structures. In general the rank singles out a distinguished noneffective spin structure, but the parity of the rank is not a quadratic form. In the case of hyperelliptic spin structures the parity is a Z/2-quadratic form. We take this as an opportunity to detail the interplay between combinatorial and algebraic rank over the moduli space. This is work in collaboration with Marco Pacini.

**Abstract**: Let \(G\) be a complex (connected) almost-simple and simply-connected group and \(C\) be a complex smooth projective curve of genus at least three. It is known that the moduli space \(\mathcal{M}_G(C)\) of semistable \(G\)-bundles over \(C\) is an irreducible projective variety. The automorphism group of \(\mathcal M_G(C)\) contains the so-called tautological automorphisms: they are induced by the automorphisms of the curve \(C\), outer automorphisms of \(G\) and tensorization by \(Z(G)\)-torsors, where \(Z(G)\) is the center of \(G\). It is a natural question to ask if they generate the entire automorphism group. Kouvidakis and Pantev gave a positive answer when \(G=SL_n\). An alternative proof has been given by Hwang and Ramanan. Later, Biswas, Gomez and Muñoz, after simplifying the proof for \(G=SL_n\), extended the result to the symplectic group \(Sp(2n)\). All the proofs rely on the study of the singular fibers of the Hitchin fibration. In this talk, we present a work in progress where, by adapting the Biswas-Gomez-Muñoz strategy, we describe the automorphism group of \(\mathcal{M}_G(C)\), for any almost-simple and simply-connected group.

**Abstract**: Fano varieties of K3 type are a special class of Fano varieties, which are usually studied for their link with hyperkaehler geometry, rationality properties, and much more. In this talk, we will recap some recent results, obtained jointly with Bernardara, Manivel, Mongardi, and Tanturri, that focus on the explicit construction of examples and the study of their Hodge-theoretical properties.

**Abstract**: What makes an intersection likely or unlikely? A simple dimension count shows that two varieties of dimension r and s are non "likely" to intersect if \(r < \text{codim } s\), unless there is some special geometrical relation among them. A series of conjectures due to Bombieri-Masser-Zannier, Zilber and Pink rely on this philosophy. After a small survey on these problems, I will present a recent application to a conjecture of Silverman on the greatest common divisor of divisibility sequences coming from certain algebraic groups. This is a joint work with F. Barroero and A. Turchet.

**Abstract**: Moduli spaces of polarized Enriques surfaces (parametrizing pairs \((S,H)\), where \(S\) is a compact, complex Enriques surface, and \(H\) is an ample line bundle on \(S\)) have several components, even if one fixes the degree of the polarization. I will present some new results determining the various irreducible components of these spaces and answering a question of Gritsenko and Hulek on the connectedness of the etale double cover from the moduli space of polarized such surfaces to the moduli space of numerically polarized such surfaces. (see https://folk.uib.no/st00895/articolimiei/Knutsen-JMPA2020-II.pdf)

**Abstract**: Moduli spaces of stable sheaves on K3 surfaces are a class of very interesting algebraic varieties, and are among the few known examples of hyperkähler varieties. In this talk I will recall the basic definitions and outline a proof, that uses the theory of Bridgeland stability, of the fact that these moduli spaces are hyperkähler varieties.

The conference "Algebraic Geometry Conference - Christmas in Roma 2021" is an Algebraic Geometry Meetting in Rome that takes place in the rione Monti. For more info here is the Link to the Official Webpage

**Abstract**: Why do some polynomial equations have only finitely many solutions in the integers? Lang-Vojta's conjecture provides a conjectural answer and relates this number-theoretic question to complex geometry. Indeed, conjecturally, a variety has only finitely many rational points if and only if it is hyperbolic. I will start out this talk explaining the Lang-Vojta conjectures, and will then present new results on dynamical systems of hyperbolic varieties, rational points on ramified covers of abelian varieties, the fundamental group of a variety covered by many pointed curves, and rigidity results for families of canonically polarised varieties. These results are mathematically independent, but all guided by the conjectures of Lang-Vojta.

**Abstract**: I will present completely algebro-geometric proofs of a theorem by T. Shiota, and of a theorem by I. Krichever, characterizing Jacobians of algebraic curves among all irreducible principally polarized abelian varieties. Shiota's characterization is in terms of the KP equation. Krichever's characterization is in terms of trisecant lines to the Kummer variety, and I will discuss only the degenerate case of his result. The proofs rely on a new theorem asserting that the base locus of a complete linear system on an abelian variety is reduced. This is a joint work with E. Arbarello and G. Pareschi.

**Abstract**: Given a smooth projective variety, its Hilbert scheme of two points often enjoys a rich structure and a beautiful geometry. The construction is very classical, but it is quite difficult to obtain explicit descriptions. In this talk we will focus on certain varieties realized as degeneracy loci, and we will provide a natural embedding of their Hilb2 in a product of grassmannians. This is a joint work in progress with E. Fatighenti, G. Mongardi and A. T. Ricolfi.

**Abstract**: The question of computing the number of maps of fixed degree d from a curve to a target variety \(X\) and verifying n incidence conditions can be viewed as a counterpart of the problem of determining the Gromov-Witten invariants of \(X\). Using degeneration and Schubert calculus, we solve this problem when the target variety is the projective space of dimension \(r\), and determine these numbers completely for linear series of arbitrary dimension when \(d\) is sufficiently large, and for all \(d\) when either \(r=1\) or \(n=r+2\). Our formulas generalize and give new proofs of very recent results of Tevelev and of Cela-Pandharipande-Schmitt. Joint work with Carl Lian.

**Abstract**: Moduli of curves play a prominent role in algebraic geometry. In particular, their rational Chow rings have been the subject of intensive research in the last forty years, since Mumford first investigated the subject. There is also a well defined notion of integral Chow ring for these objects: this is more refined, but also much harder to compute. In this talk I will present the computation of the integral Chow ring of moduli of stable 1-pointed curves of genus two, obtained by using a new approach to this type of questions (joint work with Michele Pernice and Angelo Vistoli).

**Abstract**: La congettura di Lang (1986) caratterizza le varietà complesse proiettive (o, più generalmente, Kähler compatte) iperboliche nel senso di Kobayashi come quelle di tipo generale assieme a tutte le loro sottovarietà. Lungi dall'essere dimostrata al momento, la congettura è però nota in una serie di casi paradigmatici ancorché particolari. Ci concentreremo in particolare su una direzione della congettura, spiegando come sia possibile verificare ad esempio che un quoziente libero e compatto di un dominio limitato dello spazio affine complesso abbia tutte sottovarietà di tipo generale (lavoro in collaborazione con S. Boucksom). Tempo permettendo, descriveremo alcune variazioni sul tema, considerando tipi di quozienti più generali: non più necessariamente lisci, né compatti (lavoro in collaborazione con B. Cadorel e H. Guenancia).

**Abstract**: We will describe the geometry of so-called formal-analytic arithmetic surfaces, which are an arithmetic analogue of neighborhoods of curves embedded in complex surfaces. We will study those under some positivity assumptions, proving a simple inequality that will allow us to prove an algebraization theorem that generalizes a result of Calegari-Dimitrov-Tang, as well, as bounds on fundamental groups. This is joint work with Jean-Benoît Bost.

**Abstract**: In this talk we study the moduli of irregular surfaces of general type with at worst canonical singularities satisfying \(K^2 = 4p_g-8\), for any even integer \(p_g\geq 4\). These surfaces also have unbounded irregularity \(q\). We carry out our study by investigating the deformations of the canonical morphism \(\varphi:X\to \mathbb{P}^N\), where \(\varphi\) is a quadruple Galois cover of a smooth surface of minimal degree. These canonical covers are classified into four distinct families, one of which is the easy case of a product of curves. The main objective of this talk is to study the deformations of the other three, non trivial, unbounded families. We show that, when \(X\) is general in its family, any deformation of \(\varphi\) factors through a double cover of a ruled surface and, hence, is never birational. More interestingly, we prove that, with two exceptions, a general deformation of \(\varphi\) is two--to--one onto its image, whose normalization is a ruled surface of appropriate irregularity. We also show that, with the exception of one family, the deformations of a general surface \(X\) are unobstructed even though \(H^2(T_X)\) does not vanish. Consequently, \(X\) belongs to a unique irreducible component of the Gieseker moduli space. These irreducible components are uniruled. As a result of all this, we show the existence of infinitely many moduli spaces, satisfying the strict Beauville inequality \(p_g > 2q-4\), with an irreducible component that has a proper "quadruple" sublocus where the degree of the canonical morphism jumps up. These components are above the Castelnuovo line, but nonetheless parametrize surfaces with non birational canonical morphisms. The existence of jumping subloci is a contrast with the moduli of surfaces with \(K^2 = 2p_g- 4\), studied by Horikawa. Irreducible moduli components with a jumping sublocus also present a similarity and a difference to the moduli of curves of genus \(g\geq 3\), for, like in the case of curves, the degree of the canonical morphism goes down outside a closed sublocus but, unlike in the case of curves, it is never birational. Finally, our study shows that there are infinitely many moduli spaces with an irreducible component whose general elements have non birational canonical morphism and another irreducible component whose general elements have birational canonical map.

**Abstract**: I will report on a joint work with Yuchen Liu and Taro Sano in which we construct infinitely many families of Sasaki-Einstein metrics on odd-dimensional spheres that bound parallelizable manifolds, proving in this way conjectures of Boyer-Galicki-Kollár and Collins-Székelyhidi. The construction is based on showing the K-stability of certain Fano weighted orbifold hypersurfaces.

**Abstract**: A conjecture often attributed to Kontsevich predicts that smooth projective varieties with equivalent bounded derived categories have equal Hodge numbers. In this talk, I will report on a joint work with Lombardi and Pareschi where we prove Kontsevich's conjecture for a certain class of varieties of arbitrary dimension. I will also provide an overview on the derived invariance problem, that is how much of the geometry of a variety is encoded by its derived category.

**Abstract**: I will explain some of the ideas behind the study of the birational geometry of foliations. I will illustrate then how these ideas and techniques have been used to establish the Minimal Model Program for codimension one foliations on threefolds. Features joint work with C. Spicer.

**Abstract**: The study of the bounded derived category of coherent sheaves on a smooth projective variety is a central topic in algebraic geometry. Almost all the functors arising from geometric constructions are of Fourier-Mukai type, namely they can be described by an object in the derived category of a product. In this setting, Orlov proved in 1996 that every exact fully faithful functor with adjoints is of FM type. Since then, this result has been further generalized and a useful tool is to enhance the triangulated structure on the derived category to a dg category. In this talk we consider certain admissible subcategories of the bounded derived category of cubic fourfolds, Gushel-Mukai varieties and quartic double solids, known as Kuznetsov components, and we show the strongly uniqueness of their dg enhancement making use of stability conditions with special properties. As application, we show that equivalences among the above mentioned Kuznetsov components are of FM type. This is the content of a joint work with Chunyi Li and Xiaolei Zhao.

**Abstract**: In 2004 Pachter and Speyer introduced the dissimilarity maps for phylogenetic trees and asked two important questions about their relationship with tropical Grassmannian. Multiple authors answered affirmatively the first of these questions, showing that dissimilarity vectors lie on the tropical Grassmannian, but the second question, whether the set of dissimilarity vectors forms a tropical subvariety, remained opened. In this talk, we present a negative answer to this second question. Then, we introduce a weighted variant of the dissimilarity map and show that weighted dissimilarity vectors form a tropical subvariety of the tropical Grassmannian in exactly the way that Pachter and Speyer envisioned. This tropical variety has a geometric interpretation in terms of point configurations on rational normal curves. The talk is based on a joint work with Noah Giansiracusa, Han-Bom Moon, and Luca Schaffler.

**Abstract**: Let \(X\) be an irreducible hypersurface in \(P^n\) of degree \(d\). If \(X\) has isolated singularities then \(h^i(X)= h^i(P^n)\) holds for \(i\not \in \{ n-1,n,2n-2\}\). Most hypersurfaces with isolated singularities satisfy \(h^n(X)=h^n(P^n)\). In this talk we consider hypersurfaces with semi-weighted homogeneous (e.g., ordinary multiple points or ADE-singularities) such that \(h^n(X)>h^n(P^n)\) holds. We show that if \((d,n)\) is not in an explicit finite list then the equianalytic deformation space of \(X\) is not \(T\)-smooth, i.e., this space is nonreduced or its dimension is larger than expected. A similar statement holds true for \(X\) if the \(d\)-fold cover \(Y\) of \(P^n\) ramified along \(X\) satisfies \(h^{n+1}(Y)>h^{n+1}(P^{n+1})\). This latter result generalizes classical examples of B. Segre of degree \(6m\) curves in \(P^2\) with \(6m^2\), \(7m^2\), \(8m^2\) and \(9m^2\) cusps and deformation space larger than expected.

**Abstract**: Let \(X\) be a proper smooth variety over a complete discretely valued field \(K\). One says that \(X\) has log good reduction if it admits a log smooth and proper model over the ring of integers of \(K\). In general, it is quite non-obvious whether or not such a model exists. If \(X\) is a curve, there is a neat criterion (due to T. Saito and J. Stix) in terms of the geometry of a suitable normal crossings model of \(X\). In my talk, I will present a geometric criterion for log good reduction when \(X\) is a K3 surface admitting a triple-point-free model. This is joint work in progress with J. Nicaise.

Algebraic Geometry in Roma Tre, a conference on the occasion of Sandro Verra's 70(+2)th birthday. For more info here is the Link to the Official Webpage

**Abstract**: La teoria dei blocchi conformi permette di costruire dei fibrati vettoriali su spazi di moduli di curve algebriche con notevoli proprietà geometriche e combinatorie. Questa teoria utilizza come input rappresentazioni di algebre noncommutative. Un esempio classico è dato dalle rappresentazioni di algebre di Lie affini, e i risultanti fibrati vettoriali sono stati studiati a fondo negli ultimi trenta anni, fornendo alcune straordinarie intuizioni sulla geometria degli spazi di curve. In questa conferenza, racconterò di come alcuni risultati fondamentali della teoria classica dei blocchi conformi si estendono al caso più generale consistente nel sostituire algebre di Lie affini con algebre degli operatori di vertici. In particolare, presenterò un'estensione della proprietà di fattorizzazione e nuovi esempi di teorie comologiche dei campi. Il lavoro presentato è in collaborazione con Chiara Damiolini e Angela Gibney.

Info

The seminar is usually held every **Thursday at 14:15-15:45**. Due to the pandemic the next seminars will be held in a blended format: in person with Covid-certificate (please send an email to an organizer if you want to attend) and on the platform Microsoft Teams (online).

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