Moduli Spaces
in Algebraic Geometry
and Applications
University of Campinas, July 26th - 31st
Plenary Speakers

Short Communications

Valery Alexeev
Marian Aprodu
Carolina Araujo
Agnieszka Bodzenta
Jim Bryan
Juliana Coelho
Izzet Coskun
Ben Davison
Jean Marc Drezet
Daniele Faenzi
Gavril Farkas
Daniel Greb
Victoria Hoskins
Jack Huizenga
Klaus Hulek
Emanuele Macrì
Jorge Vitório Pereira
Evangelos Routis
Giulia Saccà
Benjamin Schmidt
Alexander S. Tikhomirov
Israel Vainsencher

Aline Andrade
Ignacio Barros
Inder Kaur
Roberto Svaldi
Scientiﬁc Committee
Eduardo Esteves
Rosa Maria Miró-Roig
Giorgio Ottaviani
Ravi Vakil
Local Organizing
Committee
Marcos Jardim
Simone Marchesi

www.ime.unicamp.br/˜gtag/MSAG

Talks
1. Valery Alexeev, Compact moduli and reflection groups.
2. Marian Aprodu, Green’s conjecture and vanishing of Koszul modules.
3. Carolina Araujo, Moduli spaces of parabolic vector bundles on P1 and a special familyof Fano manifolds.
4. Agnieszka Bodzenta, Categorifying non-commutative deformation theory.
5. Jim Bryan, Banana Manifolds, Donaldson–Thomas theory, and modular forms.
6. Juliana Coelho, Hurwitz schemes and gonality of stable curves.
7. Izzet Coskun, The cohomology and birational geometry of moduli spaces of sheaves on
surfaces.
8. Ben Davison, Integrality of BPS invariants.
9. Jean-Marc Drézet, Non reduced moduli spaces of sheaves of ribbons.
10. Daniele Faenzi, Families of Cohen–Macaulay modules on singular spaces.
11. Gavril Farkas, The Prym–Green conjecture.
12. Daniel Greb, Moduli of sheaves on higher-dimensional projective manifolds.
13. Victoria Hoskins, On the motive of the stack of vector bundles on a curve.
14. Jack Huizenga, Properties of general sheaves on Hirzebruch surfaces.
15. Klaus Hulek, The cohomology of moduli spaces of cubic threefolds.
16. Emanuele Macrí, The period map for polarized hyperkähler manifolds.
17. Jorge Vitório Pereira, Irreducible components of the space of codimension one foliations.
18. Evangelos Routis, Complete complexes and spectral sequences.
19. Giulia Saccà, Remarks on degenerations of hyper-Kähler manifolds.
20. Benjamin Schmidt, Bridgeland stability and the genus of space curves.
21. Alexander S. Tikhomirov, Geography and geometry of the moduli spaces of semi-stablerank 2 sheaves on projective space.
22. Israel Vainsencher, Enumeration of singular hypersurfaces, old and new.

2

Communications
1. Aline Andrade, Orthogonal instanton bundles on Pn .
2. Ignacio Barros, Geometry of the moduli of n-pointed K3 surfaces of small genus.
3. Inder Kaur, A Torelli-type theorem for moduli spaces of semistable sheaves over nodal
curves.
4. Roberto Svaldi, On the boundedness of Calabi–Yau varieties in low dimension.

Posters
1. Charles Almeida, Moduli space of torsion free sheaves on projective spaces.
2. Alana Cavalcante, Holomorphic distributions on Fano threefolds.
3. Alberto Cazzaniga, Refined invariants for framed sheaves on P3 , and their asymptotics.
4. Douglas Guimarães, Connectedness of the punctual Hilbert and Quot schemes over theaffine space C3
5. Changho Han, A birational model of moduli of genus 4 curves using stable log surfaces
6. César Lozano Huerta, Restrictions of base loci and the Severi divisor.
7. Rick Richster, Chamber decompositions for the effective cone of Mori dream spaces.
8. Frederico Sercio, Characterizing the gonality of two-component stable curves of compact
type.
9. Helena Soares, Monads on projective varieties.
10. Jairo Menezes e Souza - Danielle Lara, On gonality, canonical models of curves andscrolls.
11. Naizhen Zhang, Multiplication maps of sections and higher rank Brill–Noether theory

Participants

1. Alex Massarenti
2. Ageu Freire
3. Amar Henni
34. Anand Deopurkar
5. André Contiero
6. Angel Carocca
7. Atoshi Chowdhury
8. Benjamin Bakker
9. Brady Ali Miliwska
10. Daniel Futata
11. Daniel Santana
12. Danilo Dias
13. Danilo Santiago
14. Ethan Cotterill
15. Ettore Turatti
16. Gabriele di Cerbo
17. João Paulo Figueredo
18. Johan Martens
19. José Eduardo Garcez
20. Juan Desimoni
21. Justin Sawon
22. Karl Christ
23. Lucas Braune
24. Luiz Carlos Sobral
25. Renan da Silva Santos
26. Renato Vidal Martins
27. Rodrigo von Flach
28. Takeshi Abe
29. Victor Pretti

4

Schedule

9h30
10h30
11h30
12h
13h
15h
16h
16h30
17h
19h

Thursday 26
Hulek
FaenziCoffee BreakBodzenta
Lunch
DrezetAndradeCoffee BreakGreb
Conference
DinnerFriday 27
AraujoAproduCoffee Break
Hoskins
Lunch
Tikhomirov
BarrosCoffee Break
Poster Session

Saturday 28
FarkasCoskunCoffee Break
Sacca
Lunch
Vainsencher
Kaur
Coffee Break
Huizenga

5

Sunday 29
Discussion
SessionBarbecueMonday 30
Alexeev
Schmidt
Coffee Break
CoelhoLunch
Bryan
SvaldiCoffee BreakRoutisTuesday 31
Macri
DavisonCoffee Break
PereiraLunch
Free
Afternoon

Abstracts of Talks
Compact moduli and reflection groups
Valery Alexeev (University of Georgia, USA)
Abstract.
I will describe a class of varieties, related to reflection groups, whose moduli spaces admit
functorial geometrically meaningful toroidal compactifications.

Green’s conjecture and vanishing of Koszul modules
Marian Aprodu (University of Bucharest, Romenia)
Abstract
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 asecond 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.
Moduli spaces of parabolic vector bundles on P1 and a special family of Fano
manifolds
Carolina Araujo (IMPA, Brazil)
Abstract
In the first part of the talk, we will introduce moduli spaces of parabolic vector bundles on P1
and describe their automorphism groups. This is a joint work of Thiago Fassarella, Inder Kaurand Alex Massarenti. In the second part of the talk, we will explore the birational geometryof these moduli spaces to describe a special family of Fano manifolds, generalizing to arbitraryeven dimension the classical double-manifestation of quartic del Pezzo surfaces: On one hand,
they are blowups of P2 at 5 general points. On the other hand, they are complete intersectionsof two quadric hypersurfaces in P4 . This is a joint work with Cinzia Casagrande.

6Categorifying non-commutative deformation theory
Agnieszka Bodzenta (University of Warsaw, Poland)
Abstract
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 topresent the deformation functor as a non-commutative Artin stack.

Banana Manifolds, Donaldson–Thomas theory, and modular forms
Jim Bryan (University of British Columbia, Canada)
Abstract
Banana Manifolds are a class of compact Calabi–Yau threefolds which are fibered by Abeliansurfaces 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 someexotic 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 fiberwiseclasses. We show that the genus g Gromov–Witten potential is an explicit genus 2 Siegel
modular form of weight 2g − 2.

Hurwitz schemes and gonality of stable curves
Juliana Coelho (Universidade Federal Fluminense, Brazil)
Abstract
In its simplest form, the Hurwitz scheme Hk,g is the moduli space of finite maps of degree kfrom a genus-g smooth curve to P1 . 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 clutchingmaps 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

The cohomology and birational geometry of moduli spaces of sheaves on surfaces
Izzet Coskun (University of Illinois at Chicago, USA)
Abstract
In the first half of this talk, I will discuss joint work with Jack Huizenga on computing thecohomology 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 isglobally generated. In the second half, I will describe joint work with Matthew Woolf on thestabilization of the cohomology of moduli spaces of sheaves on surfaces.

7Integrality of BPS invariants
Ben Davison (University of Edinburgh, Scotland)
Abstract
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 semistablesheaves defined by Joyce and Song is integer valued. This conjecture in turn is implied by thestronger 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.

Non reduced moduli spaces of sheaves of ribbons
Jean-Marc Drézet (Jussieu Mathematical Institute, France) Abstract
Some coherent sheaves on projective varieties have a non reduced versal deformation space.
For example, this is the case for most unstable rank 2 vector bundles on P2 . It may evenhappen that some moduli spaces of stable sheaves are non reduced.
Let Y be a ribbon: it is a projective variety such that C = Yred is a smooth irreducible curve,
IC2 = 0, and IC is a line bundle on C. Every coherent sheaf E on Y is, on some nonempty
open subset of Y , locally isomorphic to aOY ⊕ bOC , for some integers a, b. If this is true
on C, E is called quasi locally free. If b = 1, E is called quasi locally free of rigid type (inthis case the deformations of E have also this property). In this case, if E is stable, it is
known that the moduli space of stable sheaves M containing E is in general non reduced at
E, but Mred is smooth at E. We consider a deformation of Y to reduced projective curves
with two components, and show that a quasi locally free sheaf E can be deformed to sheaves
on the reduced curves with two components (we say that E is reachable). If E is of rigid
type, it can be deformed in two distinct ways to sheaves on the reduced curves. In particularsome components M of M deform to two components of the moduli spaces of sheaves on the
reduced curves, and M appears as the “limit” of varieties with two components, whence thenon reduced structure of M.

Families of Cohen-Macaulay modules on singular spaces
Daniele Faenzi (University of Bourgogne, France)
Abstract
The representation type of a variety expresses the complexity of the category of CohenMacaulay 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 complicatedas possible) with a special focus on singular schemes including some classes of non-normal
varieties, cones and certain reducible or non-reduced schemes

8The Prym–Green Conjecture
Gavril Farkas (Humboldt University, Germany)
Abstract
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 tothis conjecture for odd genus, recently obtained in joint work with M. Kemeny.
Moduli of sheaves on higher-dimensional projective manifolds
Daniel Greb (University of Duisburg-Essen, Germany)
Abstract
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 withrespect to stability conditions not coming from an ample line bundle.
On the motive of the stack of vector bundles on a curve
Victoria Hoskins (Freie Universität Berlin, Germany)
Abstract
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 motiveof 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 thisstack. This is all joint with Simon Pepin Lehalleur.
Properties of general sheaves on Hirzebruch surfaces
Jack Huizenga (Pennsylvania State University, USA)
Abstract
Let X be a Hirzebruch surface. Moduli spaces of semistable sheaves on X with fixed numericalinvariants 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 particulardetermine whether sheaves have the "expected"cohomology that one would naively guessfrom 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 ingredient9

is to consider the notion of prioritary sheaves, which are a slight relaxation of the notion ofsemistable sheaves which still gives an irreducible stack. Our results extend analogous resultson the projective plane by Gottsche-Hirschowitz and Bertram-Goller-Johnson to the case ofHirzerbruch surfaces. This is joint work with Izzet Coskun.

The cohomology of moduli spaces of cubic threefolds
Klaus Hulek (Leibniz Universität Hannover, Germany)
Abstract
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 toAllcock, 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 thesespaces and applications. This is joint work with S. Casalaina-Martin, S. Grushevsky and R.
Laza.

The period map for polarized hyperkähler manifolds
Emanuele Macrí (Northeastern University, USA)
Abstract
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 theperiod map is a finite union of explicit Heegner divisors that we describe. The key technicalingredient 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, andTschinkel.
As an application we will present a new short proof (by Bayer and Mongardi) for thecelebrated 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.

Irreducible components of the space of codimension one foliations
Jorge Votório Pereira (IMPA, Brazil)
Abstract
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.

10Complete Complexes and Spectral Sequences
Evangelos Routis (Kavli IPMU, the University of Tokyo, Japan)
Abstract
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 generallinear 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 ofvector 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 theset of such spectral sequences has the structure of a smooth projective variety, the ‘varietyof complete complexes’, which provides a desingularization, with normal crossings boundary,of the ‘Buchsbaum-Eisenbud variety of complexes’, i.e. a ‘wonderful compactification’ of theunion of its maximal strata.
Remarks on degenerations of hyper-Kähler manifolds
Giulia Saccà (Massachusetts Institute of Technlogy, USA)
Abstract
Using the Minimal Model Program, any degeneration of K-trivial varieties can be arranged
to be in a Kulikov type form, i.e. with trivial relative canonical divisor and mild singularities.
In the hyper-Kähler setting, we can then deduce a finiteness statement for monodromy actingon H 2 , once one knows that one component of the central fiber is not uniruled. Independentlyof this, using deep results from the geometry of hyper-Kähler manifolds, we prove that a finite monodromy projective degeneration of hyper-Kähler manifolds has a smooth filling (after
base change and birational modifications). As a consequence of these two results, we provea generalization of Huybrechts’ theorem about birational versus deformation equivalence, allowing singular central fibers. As an application, we give simple proofs for the deformation
type of certain geometric constructions of hyper-Kähler manifolds (e.g. Debarre–Voisin or
Laza–Saccà–Voisin). In a slightly different direction, we establish some basic properties (dimension and rational homology type) for the dual complex of a Kulikov type degeneration of
hyper-Kähler manifolds.
Title: Derived Categories and the Genus of Curves
Benjamin Schmidt (University of Texas at Austin, USA)
Abstract
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 other11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 curvesin the bounded derived category.

Geography and geometry of the moduli spaces of semi-stable rank 2 sheaves on
projective space
Alexander S. Tikhomirov (High School of Economics, Russia)
Abstract
We give an overview of recent results on the geography and geometry of the Gieseker–
Maruyama moduli scheme M = M (2; c1 , c2 ) of rank 2 semi-stable coherent sheaves with first
Chern class c1 = 0 or −1 and second Chern class c2 on the projective space P3 . We enumerate
all currently known irreducible components of the of M for small values of c2 , and present theconstructions of new series of components of M for arbitrary c2 . We discuss the problem ofconnectedness of M and also the problem of rationality of some series of components of M .

Enumeration of singular hypersurfaces, old and new
Israel Vainsencher (Universidade Federal de Minas Gerais, Brazil)
Abstract
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 − 1) of points isexpressed by the degree, 3(d − 1)2 , of the discriminant. Imposing a finite number of singularpoints 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.

12Abstracts of Communications
Orthogonal instanton bundles on Pn
Aline V. Andrade (Universidade Federal de Viçosa, Brazil)
Abstract
In order to obtain existence criteria for orthogonal instanton bundles on Pn , we provide abijection 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 Pn and prove that every orthogonal instanton bundle withno global sections on Pn and charge c ≥ 3 has rank (n − 1)c. We also prove that MO
Pn (c),
the coarse moduli space of orthogonal instanton bundles with no global sections on Pn , withcharge c ≥ 3 and rank (n − 1)c is affine.
Last, we construct Kronecker modules to determine the splitting type of the bundles inMO
Pn (c).
Joint work with Simone Marchesi and Rosa M. Miró-Roig.

Geometry of the moduli of n-pointed K3 surfaces of small genus
Ignacio Barros (Humboldt Universität zu Berlin, Germany)
Abstract
We prove that the moduli space of polarized K3 surfaces of genus eleven with n marked
points is unirational when n ≤ 6 and uniruled when n ≤ 7. As a consequence, we settle along standing but not proved assertion about the unirationality of M∞∞,\ for n ≤ 6. 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.

A Torelli-type theorem for moduli spaces of semistable sheaves over nodal curves
Inder Kaur (IMPA, Brazil)
Abstract
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 isisomorphic to the Jacobian of the curve. In joint work with S. Basu and A. Dan we prove asimilar statement in the case when the underlying curve is irreducible nodal.

On the boundedness of Calabi-Yau varieties in low dimension
Roberto Svaldi (University of Cambridge, England)
Abstract
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 a13

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).

14

Abstracts of Posters
Moduli space of torsion free sheaves on projective spaces
Charles Almeida (Universidade Estadual de Campinas, Brazil)
Abstract
We describe irreducible components of the moduli spaces of rank 2 torsion free sheaves on
P3 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)
Abstract
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 Pic(X) ' Z,
one obtains 18 families. Now, the objective is to complete these characterizations for any
Fano Threefold with Picard number one, for smooth hyperquadrics Qn , with n ≥ 4, and for
Grassmannians. Joint work with Maurício Corrêa.
Refined invariants for framed sheaves on P3 , and their asymptotics
Alberto Cazzaniga (AIMS-SA, Stellenbosch University, South Africa)
Abstract
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 asuperpotential. We study the generating series of a refinement of Donaldson-Thomas invariants for these moduli spaces, obtaining a closed product formula and an explicit descriptionin 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 C3
Douglas Guimarães (Universidade Estadual de Campinas, Brazil)
Abstract
We exhibit a bijection between the Quot scheme of n points over the affine space Cd and thespace of d nilpotent n by n matrices commuting with each other and satisfying a stability
15condition modulo the GLn (C) action given by conjugation. With that done, we show theirreducibility of the punctual Quot scheme over the affine space C2 , which was done also by
Baranovsky and, after that, we study the connectedness of the Quot scheme in some particularcases of d and n.
A birational model of moduli of genus 4 curves using stable log surfaces
Changho Han (Harvard University, USA)
Abstract
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 abirational moduli of genus four covers by degenerating a pair of canonical genus 4 curves anda unique quadric surface containing it, inspired by the work of Hassett. Then I will explain thegeometry 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)
Abstract
In this poster I will give examples of Mori Dream Spaces with distinct stable base locus andMori chamber decompositions of its effective cone. I will also discuss sufficient conditions forthe two decompositions to coincide. This is a work in progress with Alex Massarenti andAntonio Laface.
Characterizing the gonality of two-component stable curves of compact type
Frederico Sercio (UFF, Brazil)
Abstract
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.
Monads on projective varieties
Helena Soares (ISCTE-IUL, Portugal)
Abstract
We generalise Floystad’s theorem on the existence of monads on the projective space to alarger set of projective varieties. We consider a variety X, a line bundle L on X, and abase-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 containedin a quadric. We give necessary and sufficient conditions on integers a, b, and c for a monadof typeb
0 → (L∨ )a → OX
→ Lc → 016

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 sheafof 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 smoothprojective 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)
Abstract
Let C be an integral and projective curve; and let C 0 be its canonical model. We study the
relation between the gonality of C and the dimension of a rational normal scroll S where C 0
can lie on. We are mainly interested in the case where C is singular, or even non-Gorenstein,in which case C 0 ∼6 C. We first analyze some properties of an inclusion C 0 ⊂ S when it=
is induced by a pencil on C. Afterwards, in an opposite direction, we assume C 0 lies on acertain 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 andonly if C 0 lies on a (d − 1)-fold scroll.

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

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