Trieste Algebraic Geometry Seminar 2020–2021

Università di Trieste and SISSA run a weekly seminar on Algebraic Geometry. If you wish to give a talk,
please email me or Francesco Galuppi.

Unless specified otherwise, we meet every __Friday morning at 10:30__ on Zoom.

- Friday, May 21th at 10:30, Enrico Fatighenti (IMT Toulouse).

Title:*Fano varieties from homogeneous vector bundles*

Abstract: The idea of classifying Fano varieties using homogeneous vector bundles was behind Mukai's classification of prime Fano 3-folds. In this talk, we give a survey of some recent progress along the same lines, including a biregular rework of the non-prime Mori-Mukai 3-folds classification and some examples of higher-dimensional Fano varieties with special Hodge-theoretical properties. - Friday, May 14th at 10:30, Sara Torelli (University of Hannover).

Title:*Holomorphic one forms on moduli of curves*

Abstract: It is a classical result from the sixties that the moduli space of smooth projective curves Mg does not have non-trivial closed holomorphic 1-forms, for genus g > 2. In the talk I present a stronger result recently proven in collaboration with F. Favale and G.P. Pirola that the moduli space Mg does not have holomorphic 1-forms at all, even not closed, at least for genus g ≥ 5. Time permitting, I will then discuss further applications of the developed techniques. - Friday, May 7th at 9:30, Michail Savvas (University of California, San Diego).

Title:*Almost perfect obstruction theory and K-theoretic Donaldson-Thomas invariants*

Abstract: Perfect obstruction theories are a fundamental ingredient used to define invariants associated to moduli problems, such as virtual cycles in the Chow group and virtual structure sheaves in K-theory. However, several moduli spaces, such as the moduli space of simple perfect complexes and desingularizations of moduli stacks of semistable sheaves on Calabi-Yau threefolds, do not admit a perfect obstruction theory. In this talk, we introduce the relaxed notion of an almost perfect obstruction theory on a Deligne-Mumford stack and show that it gives rise to a virtual structure sheaf in its K-theory. This applies to many examples of interest, including the above, and enables us to define K-theoretic virtual invariants and, in particular, K-theoretic Donaldson-Thomas invariants of sheaves and complexes on Calabi-Yau threefolds. Based on joint work with Young-Hoon Kiem. - Friday, April 30th at 10:30, Kathlén Kohn (KTH Stockholm).

Title:*The adjoint of a polytope*

Abstract: This talk brings many areas together: discrete geometry, statistics, intersection theory, classical algebraic geometry, physics, and geometric modeling. First, we recall the definitions of the adjoint of a polytope given Wachspress in 1975 and by Warren in 1996 in the context of geometric modeling. They defined this polynomial to generalize barycentric coordinates from simplices to arbitrary polytopes. Secondly, we show how this polynomial appears in statistics (as the numerator of a generating function over all moments of the uniform probability distribution on a polytope), in intersection theory (as the central piece in Segre classes of monomial schemes), and in physics (when studying scattering amplitudes). Thirdly, we show that the adjoint is the unique polynomial of minimal degree which vanishes on the non-faces of a simple polytope. Finally, we observe that adjoints of polytopes are special cases of the classical notion of adjoints of divisors. Since the adjoint of a simple polytope is unique, the corresponding divisors have unique canonical curves. In the case of three-dimensional polytopes, we show that these divisors are either K3 or elliptic surfaces. This talk is based on joint works with Kristian Ranestad, Boris Shapiro and Bernd Sturmfels. - Friday, April 23th at 10:30, Fulvio Gesmundo (Max Planck Institute for Mathematics in the Sciences, Leipzig).

Title:*Geometry of tensor network varieties*

Abstract: Tensor network varieties are varieties of tensors described by the combinatorial structure of a graph. They play a major role in quantum many-body physics as well as in other areas of applied mathematics, such as algebraic complexity theory and algebraic statistics. In this seminar, I will introduce tensor network varieties, focusing on their geometric structure. I will show how to study their dimension, equations, and special subvarieties of interest. - Friday, April 16th at 10:30, Elisa Postinghel (University of Trento).

Title:*Weyl cycles on blow-ups of projective 4-spaces*

Abstract: I will introduce the definition of Weyl cycles on the blow-up X of P^n in a collection of points in general position. These are irreducible components of the intersection of pairwise orthogonal effective divisors on X that live in the Weyl orbit of exceptional divisors, where the orthogonality is taken with respect to a Dolgachev-Mukai pairing on the Picard group of X. In the case where X is a Mori dream space of dimension four or less, we can describe the geometry of and classify these objects. I will also explain how this relates to certain polynomial interpolation problems. This is joint work with M. C. Brambilla and O. Dumitrescu. - Friday, March 26th at 10:30, Fabio Tonini (University of Florence).

Title:*Cox rings and Algebraic stacks*

Abstract: In the talk I will discuss the notion of Cox ring for an algebraic stack, which extends the classical notion for varieties, via the language of torsors. I will then present some applications and possible direction of research. - Friday, March 19th at 10:30, Sergej Monavari (University of Utrecht).

Title:*Higher rank K-theoretic Donaldson-Thomas invariants of points*

Abstract: A classical way to produce (numerical) invariants is through intersection theory, usually on a smooth projective variety. We give a gentle introduction on how to use torus actions to refine invariants in several directions, for example K-theoretic and virtual invariants. As a concrete example, we explain how to extract meaningful invariants from the Quot schemes of quasi-projective smooth toric threefolds and how to refine them. We present and prove various closed formulas for different flavours of higher rank Donaldson-Thomas invariants of points, solving a series of conjectures proposed in String Theory. This is based on joint work with N. Fasola and A. Ricolfi. - Friday, March 12th at 10:30, Nicola Pagani (University of Liverpool).

Title:*Geometry of genus 1 fine compactified Jacobians*

Abstract: A classical construction in algebraic geometry associates with every nonsingular complex projective curve its Jacobian, a complex projective variety of dimension equal to the genus of the curve. A similar construction is available for singular curves, but the resulting Jacobian variety fails in general to be compact. In this talk we introduce a general abstract notion of fine compactified Jacobian for nodal curves of arbitrary genus. We focus on genus 1 and discuss combinatorial classification results for fine compactified Jacobians in the case of a single stable curve, and in the case of the universal family over the moduli space of stable pointed curves. In the former case, our abstract notion finds back objects that had already been constructed by Oda-Seshadri and others. In the latter case our complete classification exhibits new examples. We then discuss how to calculate the cohomology of these compactified Jacobians. A joint work with Orsola Tommasi. - Friday, March 5th at 10:30, Eleonora Romano (University of Genova).

Title:*Torus actions from the viewpoint of birational geometry*

Abstract: In this talk we focus on complex, smooth, projective varieties admitting a non-trivial C*-action. We introduce a new approach to investigate such varieties, using instruments coming from birational geometry. In particular, we relate small bandwidth varieties with Atiyah flips and special Cremona transformations. We report recent classification results of small bandwidth varieties, with particular attention to the case of bandwidth 3. We finally discuss some applications, in the framework of LeBrun-Salamon conjecture. This is a joint project with G. Occhetta, L. Solá Conde and J. Wiśniewski. - Friday, February 26th at 10:30, Diletta Martinelli (University of Amsterdam).

Title:*Gale duality, blowups and moduli spaces*

Abstract: The Gale correspondence provides a duality between sets of n points in projective spaces**P**^{s}and**P**^{r}when n=r+s+2. For small values of s, this duality has a remarkable geometric manifestation: the blowup of**P**^{r}at n points can be realized as a moduli space of vector bundles on the blowup of**P**^{s}at the Gale dual points. We explore this realization to describe the birational geometry of blowups of projective spaces at points in very general position. We will focus in particular on the cases where the blowup fails to be a Mori Dream Space, reporting on a joint work with Carolina Araujo, Ana-Maria Castravet and Inder Kaur. - Friday, February 19th at 10:30, Andrea Ricolfi (SISSA).

Title:*Higher rank motivic Donaldson-Thomas invariants*

Abstract: We define the Grothendieck ring of varieties and the power structure on it. We then present some formulae expressing the naive motive of the Quot scheme of points on a 3-fold, as opposed to its*virtual motive*, also known as*motivic Donaldson-Thomas invariant*. In the virtual setup, we will focus mainly on the case of affine 3-space, on which the global case is suitably modelled. The final formula for the generating function of the virtual motives fully solves the higher rank DT theory of a pair (X,F) = (3-fold,vector bundle). - Friday, February 12th at 10:30, Milena Wrobel (University of Oldenburg).

Title:*Intrinsic Grassmannians*

Abstract: Generalizing the well-known weighted Grassmanians introduced by Corti and Reid, we study certain geometric invariant theory quotients of the affine cone over the Plücker embedding of Gr(k,n). More precisely, we consider normal projective varieties whose Cox ring is defined by the Plücker ideal I_{k,n}. For k = 2, we classify the smooth Fano ones having Picard number two and give a concrete formula to compute their number for arbitrary n. - Friday, February 5th at 10:30, Andrea Petracci (Freie Universität Berlin).

Title:*On deformations and moduli of Fano varieties*

Abstract: Fano varieties are complex projective varieties with "positive curvature". They have a prominent role in algebraic geometry for many reasons, there are very few of them, they constitute the basic building blocks of algebraic varieties. In this talk I will explain how their complex structure deforms and how it is possible to construct singular points in the moduli space of K-polystable Fano varieties (i.e. Fano varieties admitting a Kähler-Einstein metric). This is joint work with Anne-Sophie Kaloghiros. - Friday, January 29th at 10:30, Kaie Kubjas (University of Aalto).

Title:*Exact solutions in low-rank approximation with zeros*

Abstract: Low-rank approximation with zeros aims to find a matrix of fixed rank and with a fixed zero pattern that minimizes the Euclidean distance to a given data matrix. We study the critical points of this optimization problem using algebraic tools. In particular, we describe special linear, affine and determinantal relations satisfied by the critical points. We also investigate the number of critical points and how the number is related to the complexity of nonnegative matrix factorization problem. This talk is based on a recent preprint together with Luca Sodomaco and Elias Tsigaridas. - Friday, January 22th at 10:30, Mateusz Michalek (University of Konstanz).

Title:*Complete quadrics: Schubert calculus for Gaussian models and semidefinite programming*

Abstract: Algebraic geometry has made great advances in the last two centuries. A particular role was played by enumerative geometry, where correct setting of moduli spaces found applications beyond mathematics. In my talk I would like to present a new work on applications of enumerative geometry in algebraic statistics. The main role will be played by the classical variety of complete quadrics and its cohomology ring. However, inspired by algebraic statistics, we will look at it from a different perspective, namely what happens when the dimension of the quadric changes. We will present two theorems, confirming conjectures posed by Nie, Ranestad, Sturmfels and Uhler. Achieving our results would not be possible without the fundamental work of De Concini, Laksov, Lascoux, Pragacz and Procesi. The talk is based on joint works with Manivel, Monin, Seynnaeve, Vodicka and Wisniewski. - Friday, January 15th at 10:30, Alex Massarenti (University of Ferrara).

Title:*On secant defectiveness of toric varieties*

Abstract: Let N be a free abelian group, M = Hom(N,Z) its dual, M_Q := M ⊗ Q the corresponding rational vector space, P ⊆ M_Q a full dimensional lattice polytope, and (X_P , H) the corresponding polarized toric variety. We study the secant defectiveness of X_P in the embedding induced by H. In particular, we give a bound, depending only on the maximum number of integer points on a facet of P, for the non secant defectiveness of X_P. Furthermore, as an application, we get an almost asymptotically sharp bound for the non secant defectiveness of Segre-Veronese varieties. - Friday, December 18th at 10:30, Emilia Mezzetti (Università di Trieste).

Title:*Togliatti systems, Weak Lefschetz Property and Galois coverings*

Abstract: In an article in collaboration with Rosa M. Mirò-Roig and Giorgio Ottaviani (Canad. J. Math. 65, 2013), we established a relation, due to apolarity, between Artinian homogeneous ideals of a polynomial ring not satisfying the Weak Lefschetz Property - WLP - and projective varieties that verify a Laplace equation of a certain order s, i.e. such that all the s-osculating spaces have dimension less than expected. Thanks to this relation, it is possible to extend to various classes of toric varieties some classical results due to Eugenio Togliatti. In the seminar, I will introduce these notions and I will speak of some recent results in collaboration with Liena Colarte and Rosa M. Miro'-Roig, relating them to Galois cyclic coverings. - Friday, December 11th at 10:30, Alex Casarotti (University of Ferrara).

Title:*Tangential weak defectiveness and generic identifiability*

Abstract: We study the uniqueness of decomposition of general tensors T as a sum of tensors of rank 1. This is done by extending the theory developed in a previous work by Mella to the framework of non-tangentially-weak-defective varieties. In particular, the infinitesimal behaviour of the tangential contact locus is linked to the birational geometry of a weakly defective but not tangentially defective variety. In this way it is possible to prove the non generic identifiability of infinitely many partially symmetric tensors. - Friday, December 4th at 10:30, Roser Homs Pons (TU München)

Title:*Primary ideals and their differential equations*

Abstract: An ideal in a polynomial ring encodes a system of linear partial differential equations with constant coefficients. Primary decomposition organizes the solutions to the PDE. This paper develops a novel structure theory for primary ideals in a polynomial ring. We characterize primary ideals in terms of PDE, punctual Hilbert schemes, relative Weyl algebras, and the join construction. Solving the PDE described by a primary ideal amounts to computing Noetherian operators in the sense of Ehrenpreis and Palamodov. We develop new algorithms for this task, and we present efficient implementations. - Friday, November 27th at 10:30, Francesco Bastianelli (Università di Bari)

Title:*Subvarieties of general hypersurfaces and their degree of irrationality*

Abstract: I will be interested in discussing various birational invariants describing how a given variety X is far from being rational. After introducing these invariants, I will focus on the case where X is a general hypersurface of large degree in the complex projective space. In particular, I will report on recent results in collaboration with C. Ciliberto, F. Flamini and P. Supino, concerning the least degree of irrationality of irreducible subvarieties of X of fixed dimension passing through a general point of X. - Friday, November 20th at 10:30, Matteo Gallet (SISSA, Trieste, and Linz)

Title:*Zero-sum cycles in flexible polyhedra*

Abstract: We show that if a polyhedron in the three-dimensional affine space with triangular faces is flexible, i.e., can be continuously deformed preserving the shape of its faces, then there is a cycle of edges whose lengths sum up to zero once suitably weighted by 1 and -1. We do this via elementary combinatorial considerations, made possible by a well-known compactification of the three-dimensional affine space as a quadric in the four-dimensional projective space. The compactification is related to the Euclidean metric, and allows us to use a simple degeneration technique that reduces the problem to its one-dimensional analogue, which is trivial to solve. This is a joint work with G. Grasegger, J. Legerský, and J. Schicho.