# MODULI SPACES IN ALGEBRAIC GEOMETRY AND APPLICATIONS

official satellite meeting of the International Congress of Mathematicians 2018, Rio de Janeiro

### Campinas, July 26th to July 31th, 2018.

Download here the poster of this event and feel free to share it!

The Algebraic Geometry community in Brazil counts with several research groups dedicated with diverse aspects of the field, like arithmetic geometry, birational geometry, classification of sheaves, holomorphic foliations, and interactions with String Theory and mathematical physics.
The theory of moduli spaces is a common thread that unites all of these areas and also has a relevant interplay with other areas of mathematics.

Our goal is to bring together senior and young researchers currently working on moduli spaces and related areas on the occasion of the 2018 International Congress of Mathematicians. We expect to create an enviroment that enhances the exchange of new ideas through discussion sessions, and helps to establish working contacts, building on the existing network that links brazilian algebraic geometers to their colleagues abroad. In this way, we hope that such meeting will highlight Brazil as a major location for work in algebraic geometry to the next generation of researchers.

### Scientific Committee

Eduardo Esteves (IMPA, Brazil)
Rosa Maria Miró-Roig (University of Barcelona, Spain)
Giorgio Ottaviani (University of Florence, Italy)
Ravi Vakil (Stanford University, USA)

### Local Organizing Committee

Marcos Jardim (University of Campinas, Brazil)
Simone Marchesi (University of Campinas, Brazil)

## Schedule

### July, 26th

9:00 a.m. : Registration

9:30 a.m. : Greb
Moduli of sheaves on higher-dimensional projective manifolds

10:30 a.m. : Faenzi
Families of Cohen-Macaulay modules on singular spaces

11:30 a.m. : Coffee Break

12:00 p.m. : Bodzenta
Categorifying non-commutative deformation theory

1:00 p.m. : Lunch

3:00 p.m. : Sawon
Lagrangian fibrations by Prym varieties

Orthogonal instanton bundles on $\mathbb{P}^n$

4:30 p.m. : Coffee Break

5:00 p.m. : Bryan
Banana Manifolds, Donaldson-Thomas theory, and modular forms

To be determined. : Social dinner

### July, 27th

10:30 a.m. : Aprodu
Green’s conjecture and vanishing of Koszul modules

11:30 a.m. : Coffee Break

12:00 p.m. : Hoskins
On the motive of the stack of vector bundles on a curve

1:00 p.m. : Lunch

4:00 p.m. : Barros
Geometry of the moduli of $n$-pointed $K3$ surfaces of small genus

4:30 p.m. : Coffee Break - Posters

5:00 p.m. : Poster Session

### July, 28th

9:30 a.m. : Farkas
The Prym-Green Conjecture

11:30 a.m. : Coffee Break

12:00 p.m. : Saccà
Remarks on degenerations of hyper-Kähler manifol

1:00 p.m. : Lunch

3:00 p.m. : Vainsencher
Enumeration of singular hypersurfaces, old and new

4:30 p.m. : Coffee Break

5:00 p.m. : Huizenga
Properties of general sheaves on Hirzebruch surfaces

### July, 29th

9:00 a.m Morning dedicated to posters and discussions

1:00 p.m. : Social Lunch

3:00 p.m. : Free afternoon

### July, 30th

9:30 a.m. : Alexeev
Compact moduli and reflection groups

10:30 a.m. : Schmidt
Bridgeland stability and the genus of space curves

11:30 a.m. : Coffee Break

12:00 p.m. : Coelho
Hurwitz schemes and gonality of stable curves

1:00 p.m. : Lunch

3:00 p.m. : Hulek
The cohomology of moduli spaces of cubic threefolds

4:00 p.m. : Svaldi
On the boundedness of Calabi-Yau varieties in low dimension

4:30 p.m. : Coffee Break

5:00 p.m. : Routis
Complete complexes and spectral sequences

### July, 31st

9:30 a.m. : Macrì
The period map for polarized hyperkähler manifolds

10:30 a.m. : Davison
Integrality of BPS invariants

11:30 a.m. : Coffee Break

12:00 p.m. : Pereira
Irreducible components of the space of codimension one foliations

1:00 p.m. : Lunch

## Plenary talks

### Valery Alexeev

#### University of Georgia, USA

Compact moduli and reflection groups
I will describe a class of varieties, related to reflection groups, whose moduli spaces admit functorial geometrically meaningful toroidal compactifications.

### Marian Aprodu

#### University of Bucharest, Romenia

Green’s conjecture and vanishing of Koszul modules
I report on a joint work in progress with G. Farkas, S. Papadima, C. Raicu and J. Weyman. Koszul modules are multi-linear algebra objects associated to an arbitrary subspace in a second exterior power. They are naturally presented as graded pieces of some Tor spaces over the dual exterior algebra. Koszul modules appear naturally in Geometric Group Theory, in relations with Alexander invariants of groups. We prove an optimal vanishing result for the Koszul modules, and we describe explicitly the locus corresponding to Koszul modules that are not of finite length. We use representation theory to connect the syzygies of rational cuspidal curves to some particular Koszul modules and we prove that our vanishing result is equivalent to the generic Green conjecture. We present some applications to Alexander invariants and other problems from Geometric Group Theory.

### Carolina Araujo

#### IMPA, Brazil

Moduli spaces of parabolic vector bundles on ${\mathbb P}^1$ and a special family of Fano manifolds
In the first part of the talk, we will introduce moduli spaces of parabolic vector bundles on ${\mathbb P}^1$ and describe their automorphism groups. This is a joint work of Thiago Fassarella, Inder Kaur and Alex Massarenti. In the second part of the talk, we will explore the birational geometry of these moduli spaces to describe a special family of Fano manifolds, generalizing to arbitrary even dimension the classical double-manifestation of quartic del Pezzo surfaces:
On one hand, they are blowups of ${\mathbb P}^2$ at 5 general points.
On the other hand, they are complete intersections of two quadric hypersurfaces in ${\mathbb P}^4$.
This is a joint work with Cinzia Casagrande.

### Agnieszka Bodzenta-Skibinska

#### University of Warsaw, Poland

Categorifying non-commutative deformation theory
We categorify non-commutative deformation theory by viewing underlying spaces of infinitesimal deformations of n objects as abelian categories with n simple objects. If the deformed collection is simple, we prove the ind-representability of the deformation functor. For an arbitrary collection we construct and ind-hull for the deformation functor. We use this hull to present the deformation functor as a non-commutative Artin stack.

### Jim Bryan

#### University of British Columbia, Canada

Banana Manifolds, Donaldson-Thomas theory, and modular forms
Banana Manifolds are a class of compact Calabi-Yau threefolds which are fibered by Abelian surfaces and have singular fibers with banana configurations: three genus zero curves meeting each of the other in two points. We construct the basic banana manifold as well as some exotic ones which are rigid and have very small Hodge numbers. We compute in closed form the Donaldson-Thomas/Gromov-Witten partition function of Banana manifolds for fiberwise classes. We show that the genus g Gromov-Witten potential is an explicit genus 2 Siegel modular form of weight 2g-2.

### Juliana Coelho

#### Fluminense Federal University, Brazil

Hurwitz schemes and gonality of stable curves
In its simplest form, the Hurwitz scheme $H_{k,g}$ is the moduli space of finite maps of degree $k$ from a genus-$g$ smooth curve to $\mathbb{P}^1$. In 1982 Harris and Mumford compactified this moduli space with the use of admissible covers. Mochizuki showed in 1995 that this compactified Hurwitz scheme is irreducible.
We introduce pointed compactified Hurwitz schemes, allowing us to consider clutching maps similar to those existing for the moduli space of stable curves. As an application, we compare the gonality of a stable curve to that of its partial normalizations. This is a joint work with Frederico Sercio.

### Izzet Coskun

#### University of Illinois at Chicago, USA

The cohomology and birational geometry of moduli spaces of sheaves on surfaces
In the first half of this talk, I will discuss joint work with Jack Huizenga on computing the cohomology of the general sheaf in moduli spaces of stable sheaves on rational surfaces. This generalizes a celebrated theorem of Göttsche and Hirschowitz. As a consequence, we classify Chern characters on Hirzebruch surfaces such that the general bundle with that character is globally generated. In the second half, I will describe joint work with Matthew Woolf on the stabilization of the cohomology of moduli spaces of sheaves on surfaces.

### Ben Davison

#### University of Edinburgh, UK

Integrality of BPS invariants
BPS numbers are certain invariants that "count" coherent sheaves on Calabi-Yau 3-folds. Because of subtleties in the definition, especially in the presence of strictly semistable sheaves, it is not a priori clear that the numbers are in fact integers. I will present a recent proof with Sven Meinhardt of this integrality conjecture. The conjecture follows from a stronger conjecture, namely that a certain constructible function on the coarse moduli space of semistable sheaves defined by Joyce and Song is integer valued. This conjecture in turn is implied by the stronger conjecture that this function is in fact the pointwise Euler characteristic of a perverse sheaf. We prove all of these conjectures by defining this perverse sheaf, and furthermore find that the hypercohomology of this sheaf, which categorifies the theory of BPS invariants, carries a natural Lie algebra structure, generalizing the theory of symmetrizable Kac-Moody algebras.

### Daniele Faenzi

#### University of Bourgogne, France

Families of Cohen-Macaulay modules on singular spaces
The representation type of a variety expresses the complexity of the category of Cohen- Macaulay modules over the associated coordinate ring. The goal of this talk is to show that almost all projective varieties are of "wild"type (meaning that this category is as complicated as possible) with a special focus on singular schemes including some classes of non-normal varieties, cones and certain reducible or non-reduced schemes

### Gavril Farkas

#### Humboldt University, Germany

The Prym-Green Conjecture
By analogy with Green’s Conjecture on syzygies of canonical curves, the Prym-Green conjecture predicts that the resolution of a general level p paracanonical curve of genus g is natural, that is, as “small” as the geometry of the curve allows. I will discuss a complete solution to this conjecture for odd genus, recently obtained in joint work with M. Kemeny.

### Daniel Greb

#### University of Duisburg-Essen, Germany

Moduli of sheaves on higher-dimensional projective manifolds
Moduli of sheaves on surfaces come in (at least) three different flavours: Gieseker moduli spaces (constructed using GIT), moduli of slope-semistable sheaves (constructed using determinant line bundles), and the Donaldson-Uhlenbeck compactification of the analytic moduli space of slope-stable vector bundles (constructed using gauge theory). In my talk, I will report on projects with various co-authors that extend the construction as well as the interplay between these different moduli spaces to higher dimensions. Special emphasis will be laid upon the construction of complex structures on gauge-theoretical moduli spaces and if time permits on the construction of Gieseker moduli spaces for sheaves that are semistable with respect to stability conditions not coming from an ample line bundle.

### Victoria Hoskins

#### Freie Universität Berlin, Germany

On the motive of the stack of vector bundles on a curve
Following Grothendieck’s vision that a motive of an algebraic variety should capture many of its cohomological invariants, Voevodsky introduced a triangulated category of motives which partially realises this idea. After describing some of the properties of this category, we explain how to define the motive of certain algebraic stacks. We show that the motive of the moduli stack of vector bundles on a smooth projective curve can be described in terms of the motive of this curve and its symmetric powers by using Quot schemes of torsion quotient sheaves. If there is time, I will also describe work in progress on proving a formula for the motive of this stack. This is all joint with Simon Pepin Lehalleur.

### Jack Huizenga

#### Pennsylvania State University, USA

Properties of general sheaves on Hirzebruch surfaces
Let $X$ be a Hirzebruch surface. Moduli spaces of semistable sheaves on $X$ with fixed numerical invariants are always irreducible by a theorem of Walter. Therefore it makes sense to ask about the properties of a general sheaf. We consider two main questions of this sort. First, the weak Brill-Noether problem seeks to compute the cohomology of a general sheaf, and in particular determine whether sheaves have the "expected" cohomology that one would naively guess from the sign of the Euler characteristic. Next, we use our solution to the weak Brill-Noether problem to determine when a general sheaf is globally generated. A key technical ingredient is to consider the notion of prioritary sheaves, which are a slight relaxation of the notion of semistable sheaves which still gives an irreducible stack.
Our results extend analogous results on the projective plane by Gottsche-Hirschowitz and Bertram-Goller-Johnson to the case of Hirzerbruch surfaces. This is joint work with Izzet Coskun.

### Klaus Hulek

#### Leibniz Universität Hannover, Germany

The cohomology of moduli spaces of cubic threefolds
The moduli space of cubic threefolds admits different compactifications, depending on the point of view one takes. These include the GIT quotient, the ball quotient model due to Allcock, Carlson and Toledo, the partial and full Kiran blow-up and the wonderful compactification. In this talk we will discuss the cohomology and the intersection cohomology of these spaces and applications. This is joint work with S. Casalaina-Martin, S. Grushevsky and R. Laza.

### Emanuele Macrì

#### Northeastern University, USA

The period map for polarized hyperkähler manifolds
The aim of the talk is to study smooth projective hyperkähler manifolds which are deformations of Hilbert schemes of points on K3 surfaces and are equipped with a polarization of fixed type. These are parametrized by a quasi-projective 20-dimensional moduli space and Verbitsky’s Torelli theorem implies that their period map is an open embedding when restricted to each irreducible component. Our main result is that the complement of the image of the period map is a finite union of explicit Heegner divisors that we describe. The key technical ingredient is the description of the nef and movable cone for projective hyperkähler manifolds (deformation equivalent to Hilbert schemes of points on K3 surfaces) by Bayer, Hassett, and Tschinkel.
As an application we will present a new short proof (by Bayer and Mongardi) for the celebrated result by Laza and Looijenga on the image of the period map for cubic fourfolds. If time permits, as second application, we will show that infinitely many Heegner divisors in a given period space have the property that their general points correspond to projective hyperkähler manifolds which are isomorphic to Hilbert schemes of points on K3 surfaces.
This is joint work with Olivier Debarre.

### Jorge Vitório Pereira

#### IMPA, Brazil

Irreducible components of the space of codimension one foliations
This talk will discuss old and new results on the structure of the moduli space of codimension one holomorphic foliations on a given projective manifold. In particular, it will be discussed how the study of deformations of rational curves along foliations can be used to describe foliations of low degree on projective spaces.

### Evangelos Routis

#### University of Tokyo, Japan

Complete Complexes and Spectral Sequences
The space of complete collineations is an important and beautiful chapter of algebraic geometry, which has its origins in the classical works of Chasles, Giambieli, Hirst, Schubert, Tyrell and others, dating back to the 19th century. It provides a ‘wonderful compactification’ (i.e. smooth with normal crossings boundary) of the space of full-rank matrices between two (fixed) vector spaces. More recently, the space of complete collineations has been studied intensively and has been used to derive groundbreaking results in diverse areas of mathematics. One such striking example is L. Lafforgue’s compactification of the stack of Drinfeld’s shtukas, which he subsequently used to prove the Langlands correspondence for the general linear group. In joint work with M. Kapranov, we look at these classical spaces from a modern perspective: a complete collineation is simply a spectral sequence of a two term complex of vector spaces. We then develop a theory involving more full-fledged (simply graded) spectral sequences of complexes of vector bundles with arbitrarily many terms. We prove that the set of such spectral sequences has the structure of a smooth projective variety, the ‘variety of complete complexes’, which provides a desingularization, with normal crossings boundary, of the ‘Buchsbaum-Eisenbud variety of complexes’, i.e. a ‘wonderful compactification’ of the union of its maximal strata.

### Giulia Saccà

#### Massachusetts Institute of Technology, USA

Remarks on degenerations of hyper-Kähler manifolds
After an overview of the role of Lagrangian fibrations in the theory of hyperkahler manifolds (HK), I will focus on the relation with O'Grady's 10 dimensional example and give some recent applications of the study of degenerations of HK manifolds to Lagrangian fibrations.

### Justin Sawon

#### University of North Carolina at Chapel Hill, USA

Lagrangian fibrations by Prym varieties
The Hitchin systems are Lagrangian fibrations on moduli spaces of Higgs bundles. Their compact counterparts are Lagrangian fibrations on compact holomorphic symplectic manifolds, such as the integrable systems of Beauville–Mukai, Debarre, Arbarello–Ferretti–Saccà, Markushevich–Tikhomirov, and Matteini. The GL-Hitchin system and the Beauville–Mukai system are both fibrations by Jacobians of curves. Thus they are isomorphic to their own dual fibrations. Moreover, Donagi–Ein–Lazarsfeld showed that the Beauville–Mukai system can be degenerated to a compactification of the GL-Hitchin system. The other Hitchin systems and the other compact integrable systems mentioned above are fibrations by Prym varieties. In this work, we explore the relations between these different Lagrangian fibrations. In particular, we describe degenerations of the compact examples to compactifications of SL-, PGL-, SO-, and Sp-Hitchin systems. We also describe dual fibrations of certain Lagrangian fibrations by Prym varieties.

### Benjamin Schmidt

#### The University of Texas at Austin, USA

Derived Categories and the Genus of Curves
A 19th century problem in algebraic geometry is to understand the relation between the genus and the degree of a curve in complex projective space. This is easy in the case of the projective plane, but becomes quite involved already in the case of three dimensional projective space. I will talk about generalizing classical results on this problem by Gruson and Peskine to other threefolds. This includes principally polarized abelian threefolds of Picard rank one and some Fano threefolds. The key technical ingredient is the study of stability of ideal sheaves of curves in the bounded derived category.

### Alexander S. Tikhomirov

#### National Research University Higher School of Economics, Russia

Geography and geometry of the moduli spaces of semi-stable rank 2 sheaves on projective space
We give an overview of recent results on the geography and geometry of the Gieseker--Maruyama moduli scheme $M=M(2;c_1,c_2)$ of rank 2 semi-stable coherent sheaves with first Chern class $c_1=0$ or $-1$ and second Chern class $c_2$ on the projective space $\mathbb{P}^3$. We enumerate all currently known irreducible components of the of $M$ for small values of $c_2$, and present the constructions of new series of components of $M$ for arbitrary $c_2$. We discuss the problem of connectedness of $M$ and also the problem of rationality of some series of components of $M$.

### Israel Vainsencher

#### Federal University of Minas Gerais, Brazil

Enumeration of singular hypersurfaces, old and new
There are 3 singular conics through 4 general points. Likewise, the number of singular plane curves of degree d passing through the appropriate number $d(d+3)/2$ of points is expressed by the degree, $3(d-1)^2$, of the discriminant. Imposing a finite number of singular points to a hypersurface in arbitrary dimension leads to polynomial formulas, some of which are explicit. The case of non-isolated singularities will also be reviewed.

## Short communications

#### Federal University of Viçosa, Brazil

Orthogonal instanton bundles on $\mathbb{P}^n$
In order to obtain existence criteria for orthogonal instanton bundles on $\mathbb{P}^n$, we provide a bijection between equivalence classes of orthogonal instanton bundles with no global sections and symmetric forms. Using such correspondence we are able to provide explicit examples of orthogonal instanton bundles with no global sections on $\mathbb{P}^n$ and prove that every orthogonal instanton bundle with no global sections on $\mathbb{P}^n$ and charge $c\geq 3$ has rank $(n-1)c$. We also prove that $\mathcal{M}_{\mathbb{P}^n}^{\mathcal{O}(c)}$, the coarse moduli space of orthogonal instanton bundles with no global sections on $\mathbb{P}^n$, with charge $c\geq 3$ and rank $(n-1)c$ is affine. Last, we construct Kronecker modules to determine the splitting type of the bundles in $\mathcal{M}_{\mathbb{P}^n}^{\mathcal{O}(c)}$. Joint work with Simone Marchesi and Rosa M. Miró-Roig.

### Ignacio Barros

#### Humboldt University, Germany

Geometry of the moduli of n-pointed $K3$ surfaces of small genus
We prove that the moduli space of polarized $K3$ surfaces of genus eleven with $n$ marked points is unirational when $n\leq 6$ and uniruled when $n\leq7$. As a consequence, we settle a long standing but not proved assertion about the unirationality of $\cal{M}_{11,n}$ for $n\leq6$. We also prove that the moduli space of polarized $K3$ surfaces of genus eleven with $9$ marked points has non-negative Kodaira dimension. We extend this to other genera.

### Inder Kaur

#### IMPA, Brazil

A Torelli-type theorem for moduli spaces of semistable sheaves over nodal curves
In 1968 Mumford and Newstead proved that the second Intermediate Jacobian of the moduli space of rank 2 semistable sheaves with fixed determinant over a smooth, projective curve is isomorphic to the Jacobian of the curve. In joint work with S. Basu and A. Dan we prove a similar statement in the case when the underlying curve is irreducible nodal.

### Roberto Svaldi

#### University of Cambridge, UK

On the boundedness of Calabi-Yau varieties in low dimension
I will discuss new results towards the birational boundedness of low-dimensional elliptic Calabi-Yau varieties, joint work with Gabriele Di Cerbo. Recent work in the minimal model program suggests that pairs with trivial log canonical class should satisfy some boundedness properties. I will show that 4-dimensional Calabi-Yau pairs which are not birational to a product are indeed log birationally bounded. This implies birational boundedness of elliptically fibered Calabi-Yau manifolds with a section, in dimension up to 5. I will explain how one could adapt our strategy to try and generalize the results in higher dimension. I will also discuss a first approach towards boundedness of rationally connected CY varieties in low dimension (joint with G. Di Cerbo, W. Chen, J. Han and, C. Jiang).

## Posters

Moduli space of torsion free sheaves on projective spaces
We describe irreducible components of the moduli spaces of rank 2 torsion free sheaves on $\mathbb{P}^3$ with prescribed singularities, and prove that the number of such irreducible components grows as the second Chern class of the sheaves grows. Additionaly, we study the case where the second Chern class of the sheaves are small, computing the exact number of irreducible components of the moduli space for particular cases. This is a joint work with Marcos Jardim, Alexander S. Tikhomirov.

Holomorphic distributions on Fano threefolds
Alana Cavalcante (Universidade Federal de Ouro Preto, Brazil)
In my Phd thesis, we studied holomorphic distributions of dimension and codimension one on smooth weighted projective complete intersection Fano threedimensional manifolds, with Picard number equal to one.
The goal of this work is to characterize those distributions whose tangent and conormal sheaves are arithmetically Cohen Macaulay (aCM), i.e. have no intermediate cohomology. In addition, we also studied the properties of their singular schemes and we construct examples of codimension one distributions on $X.$
There are more types of Fano threefolds. By considering the restriction ${\rm Pic}(X) \simeq \mathbb{Z},$ one obtains 18 families. Now, the objective is to complete these characterizations for any Fano Threefold with Picard number one, for smooth hyperquadrics $Q^n$, with $n\geq 4,$ and for Grassmannians. Joint work with Maurício Corrêa.

Refined invariants for framed sheaves on $\mathbb{P}^3$, and their asymptotics
Alberto Cazzaniga (AIMS-SA, Stellenbosch University, South Africa)
The moduli space of sheaves on the projective space framed along a plane can be realised as the moduli space of representations of the three-loop quiver with relations induced by a superpotential. We study the generating series of a refinement of Donaldson-Thomas invariants for these moduli spaces, obtaining a closed product formula and an explicit description in terms of weighted 3D-partitions. Finally, we describe the asymptotics of the generating series. Work in collaboration with Dr. D. Ralaivaosaona.

Connectedness of the punctual Hilbert and Quot schemes over $\mathbb{C}^3$
We exhibit a bijection between the Quot scheme of $n$ points over the affine space $\mathbb{C}^d$ and the space of $d$ nilpotent $n$ by $n$ matrices commuting with each other and satisfying a stability condition modulo the $GL_n(\mathbb{\mathbb{C}})$ action given by conjugation. With that done, we show the irreducibility of the punctual Quot scheme over the affine space $\mathbb{C}^2$, which was done also by Baranovsky and, after that, we study the connectedness of the Quot scheme in some particular cases of $d$ and $n$.

A birational model of moduli of genus 4 curves using stable log surfaces
Changho Han (Harvard University, USA)
Ever since the discovery of Deligne and Mumford, people considered many different ways to compactify moduli of curves. Vast majority of the known modular constructions rely on imposing singularity conditions on curves, which were shown to fit into Hassett and Keel's minimal model program on the DM compactification. Instead, I will instead showcase a birational moduli of genus four covers by degenerating a pair of canonical genus 4 curves and a unique quadric surface containing it, inspired by the work of Hassett. Then I will explain the geometry of this moduli space and state conjectures about relations to Hassett-Keel program. This is a joint work in with Anand Deopurkar.

Chamber decompositions for the effective cone of Mori dream spaces
Rick Rischter (Universidade Federal de Itajubá, Brazil)
In this poster I will give examples of Mori Dream Spaces with distinct stable base locus and Mori chamber decompositions of its effective cone. I will also discuss sufficient conditions for the two decompositions to coincide. This is a work in progress with Alex Massarenti and Antonio Laface.

Characterizing the gonality of two-component stable curves of compact type
Frederico Sercio (UFF, Brazil)
It is a well-known result that a stable curve of compact type with two components is hyperelliptic if and only if both components are hyperelliptic and the point of intersection is a Weierstrass point for each of them. We generalize this characterization for higher gonality with the use of admissible covers. This is a joint work with Juliana Coelho.

Helena Soares (ISCTE-IUL, Portugal)
We generalise Floystad's theorem on the existence of monads on the projective space to a larger set of projective varieties. We consider a variety $X$, a line bundle $L$ on $X$, and a base-point-free linear system of sections of $L$ giving a morphism to the projective space whose image is either arithmetically Cohen-Macaulay (ACM), or linearly normal and not contained in a quadric. We give necessary and sufficient conditions on integers $a$, $b$, and $c$ for a monad of type $$0 \to (L^\vee)^a \to \mathcal{O}_X^b \to L^c \to 0$$ to exist. We show that under certain conditions there exists a monad whose cohomology sheaf is simple. We furthermore characterise low-rank vector bundles that are the cohomology sheaf of some monad as above. Finally, we obtain an irreducible family of monads over the projective space and make a description on how the same method could be used on an ACM smooth projective variety $X$. We establish the existence of a coarse moduli space of low-rank vector bundles over an odd-dimensional $X$ and show that in one case this moduli space is irreducible. Joint work with Simone Marchesi and Pedro Macias Marques.

On gonality, canonical models of curves and scrolls
Danielle Lara (Universidade Federal de Viçosa, Brazil)
Jairo Menezes e Souza (Universidade Federal de Goiás, Brazil)
Let $C$ be an integral and projective curve; and let $C'$ be its canonical model. We study the relation between the gonality of $C$ and the dimension of a rational normal scroll $S$ where $C'$ can lie on. We are mainly interested in the case where $C$ is singular, or even non-Gorenstein, in which case $C'\not\cong C$. We first analyze some properties of an inclusion $C'\subset S$ when it is induced by a pencil on $C$. Afterwards, in an opposite direction, we assume $C'$ lies on a certain scroll, and check some properties $C$ may satisfy, such as gonality and the kind of its singularities. At the end, we prove that a rational monomial curve $C$ has gonality $d$ if and only if $C'$ lies on a $(d-1)$-fold scroll.

Multiplication Maps of sections and higher rank Brill-Noether Theory
Naizhen Zhang (Katholieke Universiteit Leuven, Belgium)
We discuss our recent progress on higher rank Brill-Noether Theory through the lens of multiplication maps of sections of line bundles on smooth curves.

## Travel Information

Accomodation
The organization has a special deal with the Hotel Dan Inn Cambui, which is located in the Cambui neighbourhood downtown Campinas.
You can write to either one of the local organizers, until June 30th, with the details of your staying and we will inform the hotel with your request. If you do so, please specify if you want a single room or if you would like to share one with another participant.

Visa and permanence

Who needs a visa to visit Brazil?

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Entry visas for visiting Brazil are currently required from citizens from Australia, Canada, Haiti, the United States of America, most countries in Africa, Asia and Oceania, and a few other countries. Check the brazilian consulate in your country of birth to check if you need a visa.
The whole process of requiring a vista is the sole responsibility of each individual participant.

Letter of invitation
Brazilian authorities may request letters of invitation from the MSAG participants. Contact us if you need such letter.

Airports

GRU - Guarulhos International Airport
Hosts domestic and international flights.
Website of the airport

GRU Airport (Cumbica) connects Guarulhos to the other main cities, including Campinas. There is the following direct executive bus from the airport to Campinas Bus Station that should be taken at Terminal 2:

Executive Line
Company: VB Transportation (Lira Bus)
Bus fare to Campinas Bus Station: Reais 36,00
Time schedule:
Cumbica to Campinas (Lira bus): 6h30, 8h00, 9h00, 10h30, 12h00, 14h00, 16h00, 17h30, 18h45, 20h00, 21h30, 23h00, 00h30.

From the Bus Station to Barão Geraldo District (where the School/Workshop Venue is located) it is recommended to take a taxi (see Taxi and related services).

CGH - São Paulo/Congonhas Airport
Domestic flights.
Website of the airport
CGH Airport connects São Paulo to the other main cities, including Campinas. There is the following direct executive bus from the airport to Campinas Bus Station:

Executive Line
Company: VB Transportation (Lira Bus)
Bus fare to Campinas Bus Station: Reais 31,00
Time schedule:
Congonhas to Campinas (Lira Bus): 7h00, 9h00, 12h00, 14h30, 15h00, 17h30, 20h30, 23h15.

From the Bus Station to Barão Geraldo District (where the School/Workshop Venue is located) it is recommended to take a taxi (see Taxi and related services).

VCP - Viracopos International Airport
Domestic and (some) international flights.
Website of the airport
Viracopos International Airport is connected to the major cities in the metropolitan area of Campinas via public transportation (193 from the Airport to Bus Station and 331 from the Bus Station to Terminal Barão Geraldo, R$9,40) or the following executive line: Executive Line Company: VB Transportation (Lira Bus) Bus fare to Campinas Bus Station: Reais 13,00 Time schedule: Viracopos to Campinas: 05h45, 06h50, 08h00, 09h00, 10h15, 11h15, 13h00, 14h30, 16h00, 17h30, 19h00, 20h15, 21h45, 23h15, 00h30. From the Bus Station to Barão Geraldo District (where the School/Workshop Venue is located) it is recommended to take a taxi (see Taxi and related services). General Information Time zone The time zone for Campinas is GMT - 3:00. Climate Campinas has a tropical climate. Winter (June to September) typically has temperatures in the 11ºC (51.8ºF)– 25ºC (77ºF). Currency The official currency of Brazil is the Real, plural reais (BRL or R$). The exchange of money and traveler checks can be done at banks and travel agencies.

Mains eletricity
The standard voltage in Campinas is 110 - 120 volts. Some buildings (including several hotels) have additional 220-volt power plugs, as a rule clearly marked. The current standard for power outlets in Brazil looks like this (please do not ask us why):

There are few of these outlets in our building though. We shall be able to provide a limited number of adapters (depending on how many we can scratch from our drawers). In our building one can find many outlets suitable for the US standard, and also for the European one (two round sticks).

Taxis and related services
Regular cabs offer rides charged by the meter, except for certain rides starting from local airports and most taxi drivers only accept direct payment in cash. Mobile apps such as 99 Taxi, Uber and Cabify allow the use payment via preregistered credit card.

Phone Service
The list of city codes, operator codes and country codes can be requested in the reception of a hotel.
Area Code for Campinas: 19
Country Code for Brazil: 55
National Long Distance Calls: 0 + operator code + city code + phone number
National Long Distance Collect Calls: 90 + operator code + city code + phone number
International Long Distance Calls: 00 + operator code + country code + city code + phone number
0800 Calls: Call the desired number

Emergency Numbers
Police: 190
First Aid: 192
Fire Department: 193