Workshop on Algebraic Geometry and Physics 2025. Moduli Spaces in (Super)Geometry and Mathematical Physics. A celebration of Ugo Bruzzo's 70th birthday

6 Oct 2025, 09:30 → 10 Oct 2025, 19:00 America/Sao_Paulo

Littoral Hotel

WAGP2025 marks the 20th edition of a distinguished series of Schools and Workshops organized by the Mathematical Physics and Geometry group at the International School for Advanced Studies in Trieste, in collaboration with leading institutions worldwide. Previous events have been held in cities such as Trieste, Como, and Genova (Italy), Salamanca and Medina del Campo (Spain), Luminy and St. Jean de Monts (France), Lisbon (Portugal), Philadelphia (USA), Vienna (Austria), Seoul (Korea), Maresias (Brazil), Tianjin and Hangzhou (China), Leiden (Netherlands), and Guanajuato (Mexico).

 

This year’s theme will be:

Moduli Spaces in (Super)Geometry and Mathematical Physics

 

The workshop will also provide a special opportunity to celebrate Ugo Bruzzo’s 70th birthday and honor his substantial contributions to algebraic geometry and mathematical physics.

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Speakers

Gessica Alecci (Politecnico di Torino, Italy)

Charles Almeida (Federal University of Minas Gerais, Brazil)

Carolina Araujo (Instituto de Matemática Pura e Aplicada, Brazil)

Fábio Arceu Ferreira (Federal University of Paraíba, Brazil)

Giordano Cotti (Universidade de Lisboa, Portugal)

Peter Dalakov (American University in Bulgaria, Bulgaria)

Pedro Henrique Dos Santos (University of Campinas, Brazil)

Rodrigo Gondim (Federal University of Pernambuco, Brazil)

Michele Graffeo (SISSA, Italy)

Beatriz Graña Otero (Universidad de Salamanca, Spain)

Rafael Holanda (Federal University of Pernambuco, Brazil)

Daniel Hernández Ruipérez (Universidad de Salamanca, Spain)

Gianni Landi (Università degli Studi di Trieste, Italy)

Valeriano Lanza (Universidade Federal Fluminense, Brazil)

William Montoya (Universide de Campinas, Brazil)

Leonardo Roa Leguizamon (Universide de Campinas, Brazil)

Arpan Saha (University of Campinas, Brazil)

Renato Vidal Martins (Universide de Campinas, Brazil)
 

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Organizers

Marcos Jardim (University of Campinas, Brazil)

Emanuele Macrì (Université de Paris-Saclay, France)

Andrea Ricolfi (SISSA, Italy)

Francesco Sala (Università di Pisa, Italy)

Monday, 6 October

09:30 → 10:30 Moduli of stable super maps

We construct the moduli of stable supermaps as an algebraic superstack with superschematic and separated diagonal. We prove that its bosonic reduction is an affine linear scheme over the stack of stable spin maps, so that it is not proper except in a few particular cases. We also compute the virtual dimension of the moduli superstack of stable supermaps, and prove that it coincides with previous calculations when the target of the supermaps is bosonic.

Speaker: Daniel Hernández Ruipérez (Universidad de Salamanca, Spain)

10:30 → 11:00 Coffee break

11:00 → 12:00 TBA

TBA

Speaker: Valeriano Lanza (Universidade Federal Fluminense, Brazil)

12:00 → 15:00 Lunch Break

15:00 → 16:00 Higgs Grassmannians

We consider a Higgs bundle $(E, \phi)$. Its Higgs Grassmiannans are subschemes of the usual Grassmannian bundles of $E$ that parameterise Higgs quotients of $(E, \phi)$. We recall how to define them, present some results about their structure, and explain how they can be used to prove some results about Higgs bundles satisfying a strong semistability condition.

Speaker: Beatriz Graña Otero (Universidad de Salamanca, Spain)

16:00 → 16:30 Coffee break

16:30 → 17:30 On the Quot scheme of a fat point

In this project we investigate the geometry of the scheme $\mathrm{Quot}_{\mathcal{E}}^d X$, where $X$ is a smooth quasi-projective variety, $d$ is a non-negative integer and $\mathcal{E}\in\mathrm{Coh} X$ is the ideal sheaf of a fat point. We mainly focus on powers of the maximal ideal sheaf and complete intersections. Along the way we provide generating series for some motivic invariants attached to these schemes. The main techniques we adopt come from Commutative Algebra. Using the notion of Apolarity and Hilbert--Samuel functions we recover the Bia{\l{}}ynicki-Birula decomposition of the Quot schemes. This is a work in progress with Michele Graffeo (SISSA), aiming to begin a systematic study of these schemes in the non-locally free setting.

Speaker: Pedro H. dos Santos (Universidade Federal de Pernambuco (UFPE), Brazil)

Tuesday, 7 October

09:30 → 10:30 TBA

TBA

Speaker: Giovanni Landi (Università degli Studi di Trieste, Italy)

10:30 → 11:00 Coffee break

11:00 → 12:00 On the Noether-Lefschetz theory in projective toric orbifolds

In 2012, Bruzzo and Grassi proved a Noether-Lefschetz theorem for toric varieties, which claims that for a (2k+1)-dimensional projective toric orbifold with suitable conditions on a very general quasi-smooth hypersurface $X$, each $(k,k)$-cohomology class on $X$ comes from the ambient toric variety. The Noether-Lefschetz locus is the locus of quasi-smooth hypersurfaces with the same degree such that there exists a $(k,k)$-cohomology class that does not come from the ambient toric variety. In this talk, I will present the main results about the Noether-Lefschetz loci in toric varieties based on my joint work with Prof. Ugo Bruzzo in recent years.

Speaker: William D. Montoya (Universidade Estadual de Campinas, Brazil)

12:00 → 15:00 Lunch break

15:00 → 16:00 Automorphisms of quartic surfaces and Cremona transformations

In this talk, I will address the following question, attributed to Gizatullin: ``Which automorphisms of a smooth quartic surface in projective 3-space are restrictions of Cremona transformations of the ambient space?'' Corti and Kaloghiros have introduced a general framework that is extremely useful for approaching this problem, namely, a special version of the Sarkisov program for Calabi-Yau pairs. I will report on recent progress on Gizatullin’s problem obtained using this theory, in collaborations with Alessio Corti and Alex Massarenti, and with Daniela Paiva and Sokratis Zikas.

Speaker: Carolina Araujo (IMPA, Brazil)

16:00 → 16:30 Coffee break

16:30 → 17:30 TBA

TBA

Speaker: Rodrigo Gondim (Federal University of Pernambuco, Brazil)

Wednesday, 8 October

09:30 → 10:30 TBA

TBA

Speaker: Giordano Cotti (Universidade de Lisboa, Portugal)

10:30 → 11:00 Coffee break

11:00 → 12:00 Splitting of supervector bundles on projective superspaces

In this talk, we will introduce basic concepts of superalgebraic geometry and explore why certain classical foundational results, such as the Birkhoff-Grothendieck splitting criterion, do not extend naturally to the supergeometric setting. We then present a splitting criterion for supervector bundles and give some examples of supervector bundles with vanishing cohomology that do not split.

Joint work with Ugo Bruzzo (SISSA).

Speaker: Charles Almeida (Universidade Federal de Minas Gerais, Brazil)

12:00 → 15:00 Lunch break

15:00 → 19:00 Free afternoon

19:00 → 21:00 Conference dinner

Thursday, 9 October

09:30 → 10:30 TBA

TBA

Speaker: Renato Vidal Martins (Universide de Campinas, Brazil)

10:30 → 11:00 Coffee break

11:00 → 12:00 The Hilbert scheme of points and its motive

The Hilbert scheme of points on a quasi-projective variety is a classical object in algebraic geometry. However, its geometry is nowadays still not completely accessible. On the other hand, the motive of a variety $X$ is an invariant attached to $X$ carrying a lot of information about its geometry, and it is considered as a universal Euler characteristic. In a joint project with Monavari, Moschetti and Ricolfi we give general formulas to compute the motive of the Hilbert scheme of points, provided the knowledge of a finite amount of data (that we give explicitly in some cases). In my seminar I will present our formulas and I will show many applications.

Speaker: Michele Graffeo (Sissa, Italy)

12:00 → 15:00 Lunch break

15:00 → 16:00 TBA

TBA

Speaker: Gessica Alecci (Politecnico di Torino, Italy)

16:00 → 17:00 Coffee break

17:00 → 18:00 On the exceptional set of crepant resolutions of abelian singularities

Let $G$ be a finite abelian subgroup of $\mathsf{SL}(n, \mathbb{C})$, and suppose there exists a toric crepant resolution $ \phi: X \longrightarrow \mathbb{C}^n / G$ of the quotient variety $\mathbb{C}^n / G$. Let $\mathsf{Exc}(\phi) = E_1 \cup \dots \cup E_s$ be the decomposition of the exceptional set of $\phi$ into irreducible components. In this seminar, I will show that for every $i$ there exists an open subset $U_i$ of $X$ such that $E_i \subset U_i$, and $U_i$ is isomorphic to the total space of the canonical bundle $\omega_{E_i}$ of $E_i$. Furthermore, $X = U_1 \cup \dots \cup U_s$. This contributes to the collection of results aimed at solving a classical problem, i.e., to determine which submanifolds of a complex manifold have a neighborhood isomorphic to a neighborhood of the zero section of their normal bundle.

Speaker: Fábio Arceu Ferreira (Federal University of Rio Grande do Norte, Brazil)

Friday, 10 October

09:30 → 10:30 Towards an algebraic proof for the Codimension One Theorem

In 1996, David Cox proved that for a given projective toric variety of dimension $n$, its homogeneous coordinate ring modulo $n+1$ forms with the same ample degree, that do not vanish simultaneously, must have dimension one in the component of the critical degree of the forms.

This result, known as the Codimension One Theorem, was generalized by Cattani-Cox-Dickenstein and even further by Cox-Dickenstein. We will discuss these generalizations and their geometric ingredients involved in their proofs. We will conclude with an algebraic proof of Cox's theorem in the case of a product of projective spaces.

Speaker: Rafael Holanda (Universidade Federal de Pernambuco, Brazil)

10:30 → 11:00 Coffee break

11:00 → 12:00 Non-Kähler Hodge--Lefschetz theory and the Bianchi identity

Being Kähler imposes severe constraints on the cohomology of compact complex manifolds such as the Hard Lefschetz property, and the question of how far this generalises beyond the class of Kähler manifolds has been of great interest for a while. In this talk, I shall report on ongoing joint work with Mario García Fernández and Raúl González Molina that abstracts out the definition of a variation of Hodge--Lefschetz structure and provides evidence that, under certain natural assumptions, such a structure exists more generally on distinguished subspaces within moduli spaces of Bismut--Ricci-flat metrics that are pluriclosed up to source terms. In particular, these distinguished subspaces may be regarded as replacements for the Kähler cone, with affine structure modelled on a subspace of the (1,1) Aeppli cohomology of the compact complex manifold.

Speaker: Arpan Saha (Universidade Estadual de Campinas, Brazil)

12:00 → 15:00 Lunch break

15:00 → 16:00 Coherent Systems on Surfaces

Let $X$ be a smooth, irreducible, projective surface. A coherent system on $X$ is a pair $(E, V)$ where $E$ is a coherent sheaf on $X$ and $V$ is a finite-dimensional vector space. Associated to coherent systems there is a notion of stability that depends on a parameter $\alpha \in \mathbb{Q}[m]$. In this talk, we describe the moduli space of coherent systems for $\alpha \gg 0$, present topological and geometric properties of this moduli space, and describe the structure of chambers and walls for coherent systems on the projective plane when $\dim(V) = 2$.
This is joint work with L. Costa, I. Macías-Tarrío, and a joint work with O. Mata-Gutiérrez, and H. Torres-López.

Speaker: Leonardo Roa-Leguizamón (Universidade Estadual de Campinas, Brazil)

16:00 → 16:30 Coffee break

16:30 → 17:30 TBA

TBA

Speaker: Peter Dalakov (American University in Bulgaria, Bulgaria)