Arithmetic, Birational Geometry,
and Moduli Spaces

Poster Titles and Abstracts

Alexander Betts, Harvard University
$p$-adic obstructions and the Section Conjecture
"Grothendieck's Section Conjecture predicts that the set of rational points on a curve of genus at least two should be in bijection with the set of conjugacy classes of splittings of a certain sequence of etale fundamental groups. This set of splittings -- known as the section set -- is highly mysterious, and consequently the Section Conjecture is completely unknown outside a handful of examples.
This poster will survey two new techniques for studying the locally geometric part of the section set, united under the rough slogan that ""$p$-adic obstructions to rational points can be used to study locally geometric sections"". In the first, developed jointly with Jakob Stix, we explain how the proof of the Mordell Conjecture by Lawrence--Venkatesh can be adapted into a general finiteness theorem for the locally geometric section set. In the second, developed jointly with Martin Lüdtke and Theresa Kumpitsch, we explain how the Chabauty--Kim method can be used to compute the locally geometric section set in some examples. Taken together, this suggests that $p$-adic obstructions provide a fruitful new avenue for the study of the Section Conjecture."

Barbara Bolognese, University of Frankfurt
P=W phenomena on abelian varieties
Let X be a complex abelian variety. We prove an analogue of both the (cohomological) P = W conjecture and the geometric P = W conjecture connecting the finer topological structure of the Dolbeault moduli space of topologically trivial semistable Higgs bundles on X and the Betti moduli space of characters of the fundamental group of X. The geometric heart of our approach is the spectral data morphism for Dolbeault moduli spaces on abelian varieties that naturally factors the Hitchin morphism and whose target is not an affine space of pluricanonical sections, but a suitable symmetric product.

Sebastian Bozlee, Tufts University
On compactifications of M_g,n with colliding markings
"One way of forming modular compactifications of the space of n-pointed smooth algebraic curves is by allowing marked points to collide, as in the spaces of weighted pointed stable curves constructed by Hassett. This paper studies all such ways of constructing compactifications by allowing markings to collide.
We find that for any modular compactification, collisions of markings are controlled by a simplicial complex which we call the collision complex. Conversely, we identify modular compactifications with essentially arbitrary collision complexes, including complexes not associated to any space of weighted pointed stable curves. These moduli spaces classify the modular compactifications by nodal curves with smooth markings as well as the modular compactifications in genus one with Gorenstein curves and smooth markings"

Andrei Bud, Goethe-Universität Frankfurt
The Prym-Brill-Noether divisor
Prym-Brill-Noether theory is concerned with the study of Brill-Noether loci on Prym curves that take the Prym structure into account. We approach this topic from the perspective of Moduli Theory: we compute the class of a Prym-Brill-Noether divisor and show irreducibility of a universal Prym-Brill-Noether locus. 

George Cooper, University of Oxford
Moduli Spaces of Hyperplanar Admissible Flags in Projective Space
We prove the existence of quasi-projective coarse moduli spaces parametrising certain complete flags of subschemes of a fixed projective space ℙ(V) up to projective automorphisms. The flags of subschemes being parametrised are obtained by intersecting non-degenerate subvarieties of ℙ(V) of dimension n by flags of linear subspaces of ℙ(V) of length n, with each positive dimension component of the flags being required to be non-singular and non-degenerate, and with the dimension 0 components being required to satisfy a Chow stability condition. These moduli spaces are constructed using non-reductive Geometric Invariant Theory for actions of groups whose unipotent radical is graded, making use of a non-reductive analogue of quotienting-in-stages developed by Hoskins and Jackson.

Fernando Figueroa, Princeton University
Fundamental groups of low-dimensional log canonical singularities
Over the complex numbers, the fundamental group of an isolated singularity can be defined as the fundamental group of a sufficiently small punctured neighborhood of an algebraic singularity. We study the fundamental groups of isolated log canonical singularities of dimension at most 4. In dimension  3, we show that every surface group appears as the fundamental group of a 3-fold log canonical singularity. In contrast, we show that for $r \geq 2$ the free group on $r$ generators is not the fundamental group of a 3-dimensional lc singularity. In dimension 4, we show that the fundamental group of any 3-manifold smoothly embedded in $\mathbb{R}^4$ is the fundamental group of an isolated lc singularity. In particular, every free group is the fundamental group of a log canonical singularity of dimension 4. This is based on joint work with Joaquín Moraga.

Deniz Genlik, The Ohio State University
Holomorphic Anomaly Equations For C^n/Z_n
Physics approach to higher genus mirror symmetry predicts that Gromov-Witten potential of a Calabi-Yau threefold should satisfy certain partial differential equations; namely, the holomorphic anomaly equations. Recently, by works of Lho-Pandharipande, these equations are mathematically proved for some Calabi-Yau threefolds including C^3/Z_3. After their works, many others proved holomorphic anomaly equations for various three-dimensional targets. We generalized the work of Lho-Pandharipande on C^3/Z_3 and proved holomorphic anomaly equations for C^n/Z_n for n greater than or equal to 3, which is a result beyond the consideration of physicists. This is a joint work with Hsian-Hua Tseng.

Andreas Gross, Goethe University Frankfurt
Vector bundles in tropical geometry
Although tropical vector bundles have been introduced by Allermann ten years ago, very little has been said about their structure and their relation to vector bundles on algebraic varieties. My poster will sketch recent progress in joint work with Martin Ulirsch and Dmitry Zakharov, and Inder Kaur, Martin Ulirsch, and Annette Werner. Highlights of the theory are a tropical analogue of the Weil-Riemann-Roch theorem and the fact that the non-Archimedean skeleton of the moduli space of semihomogeneous vector bundles on a totally degenerate Abelian variety is isomorphic to a suitably defined moduli space of tropical semihomogeneous bundles. 

Siddarth Kannan, UCLA
Weight zero cohomology of moduli of hyperelliptic curves
I will discuss the moduli space H_g,n of n-pointed hyperelliptic curves of genus g, and how to use moduli spaces of pointed admissible G-covers to construct a normal crossings compactification of this space. Deligne’s weight spectral sequence identifies the reduced homology of the dual complex of the boundary divisor with the weight zero compactly supported cohomology of H_g,n. This allows us to give a sum-over-graphs formula for the weight zero compactly supported Euler characteristic, as a virtual representation of the symmetric group. This poster is based on joint work with Madeline Brandt and Melody Chan.

Kevin Kuehn, Goethe University Frankfurt
Functorial Tropicalization of Logarithmic Schemes over Valuation Rings
We provide a framework to construct a tropicalization map for a logarithmic scheme over a valuation ring with rank 1 valuation. For that, we establish a category of abstract polyhedra and compare it to a subcategory of monoids extending the stalk of the ghost sheaf of the closed point. This stalk is given by the monoid of non-negative elements of the value group and we require a certain finiteness condition for any monoid extending this. Our work naturally extends the well-understood case of trivially valued fields, where the base monoid is trivial and all involved polyhedra are cones.

Arne Kuhrs, Goethe University Frankfurt
Buildings, valuated matroids, and tropical linear spaces
Affine Bruhat-Tits buildings are geometric spaces extracting the combinatorics of algebraic groups. The building of PGL parametrizes flags of subspaces/lattices in or, equivalently, norms on a fixed finite-dimensional vector space, up to homothety. It has first been studied by Goldman and Iwahori as a piecewise-linear analogue of symmetric spaces. The space of seminorms compactifies the space of norms and admits a natural surjective restriction map from the Berkovich analytification of projective space. Inspired by Payne's result that analytification is the limit of all tropicalizations, we show that the space of seminorms is the limit of all tropicalized linear subspaces of rank r (as the embedding and the dimension of the ambient projective space vary), and prove a faithful tropicalization result for compactified linear spaces. The space of seminorms is in fact the tropicalization of the universal realizable valuated matroid, extending a result of Dress and Terhalle.  This is joint work with Luca Battistella, Kevin Kühn, Martin Ulirsch and Alejandro Vargas.

Jennifer Li, Princeton University
On the cone conjecture for log Calabi-Yau threefolds
Let $(Y, D)$ be a log Calabi-Yau threefold, meaning that $Y$ is a smooth projective threefold over $\mathbb{C}$ and $D \subset Y$ is a normal crossing divisor such that $K_{Y}+D$ is trivial. Moreover, suppose that $D$ is maximal, meaning there exists a $0$-stratum of $D$. Suppose there exists a $K3$-fibration $f: (Y, D) \rightarrow (\mathbb{P}^{1}, \infty)$ with $D = f^{\ast}(\infty)$ and $H^{3}(Y)=0$. Such fibrations arise as mirrors to Fano threefolds. We show that the pseudoautomorphism group of $Y$ acts on the codimension one faces of the movable effective cone of $Y$ with finitely many orbits. This is implied by the Kawamata-Morrison-Totaro cone conjecture.

Yuze Luan, UC Davis
Irreducible components of Hilbert scheme of points on non-reduced curves
We classify the irreducible components of the Hilbert scheme of n points on non-reduced algebraic planar curves, and give a formula for the multiplicities of the irreducible components. The irreducible components are indexed by partitions of n; all have dimension n; and their multiplicities are given as a polynomial of the parts of the corresponding partitions.

Erik Paemurru, IMSA-Miami, ICMS-Sofia
Log canonical thresholds of high multiplicity plane curves
We classify log canonical thresholds at points of multiplicity $d-1$ for plane curves of degree $d$. As a consequence, we describe all possible values of log canonical threshold that are less than $2/(d-1)$ for plane curves of degree $d$. In addition, we compute log canonical thresholds for all plane curves of degree less than $6$.

Weite Pi, Yale University
Mumford relations for the moduli of one-dimensional sheaves on P^2
The original Mumford relations are certain relations in the cohomology ring of the moduli of semistable vector bundles over a fixed Riemann surface. In a 1992 paper, Kirwan proved that the Mumford relations are complete for the rank two case, as predicted by Mumford himself. This fails for higher ranks, but can be fixed by introducing additional ‘generalized’ Mumford relations. We present work on formulating the Mumford and generalized Mumford relations for an analogous moduli space, the moduli of one-dimensional sheaves on the projective plane. As applications, we prove a minimal generation result and a cohomological χ-dependence result, and explicitly determine some cohomology rings in low degrees. Based on joint work with Y. Kononov, W. Lim, and M. Moreira, and J. Shen.

Felix Röhrle, University of Tübingen
Topology of the moduli space of tropical $\mathbb{Z}/p\mathbb{Z}$-covers
"In recent work, Chan--Galatius--Payne have developed tools to study the topology of tropical moduli spaces. The original motivation was the application to the moduli space of (marked) curves, in which case the tropical computation gave new insights into certain graded pieces of the cohomology of the algebraic moduli space $\mathcal{M}_{g,n}$. This transfer of information from tropical to algebraic moduli space was made possible by the work of Abramovich--Caporaso--Payne.
In our current project, we use the tools of Chan--Galatius--Payne and apply them to the moduli space $M_{g, \mathbb{Z}/p\mathbb{Z}}^{trop}$ of cyclic unramified tropical covers with Galois group $\mathbb{Z}/p\mathbb{Z}$. These spaces are similar in behavior but mildly more complicated than $M_{g,n}^{trop}$. We give some general results (i.e. we describe a fairly large locus in the moduli space which turns out to be contractible), but we also give some explicit computations of the homotopy type of the moduli space in genus $g = 2$. Our findings are motivated by work in progress by Pedro Souza, which promises application to the cohomology of algebraic moduli spaces. This is joint work in progress with Yassine El Maazouz, Paul Alexander Helminck, Pedro Souza, and Claudia Yun."

Nolan Schock, University of Illinois, Chicago
Quasilinear tropical compactifications
A linear tropical compactification is a tropical compactification of a complement of a hyperplane arrangement. A number of remarkable properties of these compactifications can be seen purely from the combinatorial side. Namely, any stratum of a linear tropical compactification is also linear, and the Chow ring of a linear tropical compactification is the same as the tropical Chow ring of the corresponding tropical fan. This perspective has led to a number of important developments in combinatorial Hodge theory. We introduce a larger class of tropical compactifications, called quasilinear tropical compactifications, which satisfy generalizations of these remarkable properties. As applications, we describe the Chow rings of certain compactifications of moduli spaces of hyperplane arrangements and marked del Pezzo surfaces.

Pedro Souza, Frankfurt University
The skeleton of the moduli space of cyclic covers of curves
In this work, we show that the skeleton of the Hurwitz moduli space of cyclic covers is isomorphic to the moduli space of tropical Hurwitz cyclic covers. We combine this with results of Chan-Galatius-Payne to find non-zero rational cohomology classes of the Hurwitz moduli space of cyclic covers.

Talon Stark, UCLA
The cone conjecture in relative dimension 2
We prove that, for a klt CY pair (X/S,\Delta) of relative dimension two, there are finitely many PsAut(X/S,\Delta)-orbits of Mori chambers and of faces of Mori chambers, where PsAut(X/S,\Delta) are the birational automorphisms of X over S that are isomorphisms in codimension 1 and preserve \Delta. This proves the geometric statement that there are finitely many isomorphism classes of small Q-factorial modifications (equivalently, for X terminal, minimal models) and rational contractions of X over S. Joint with Joaquín Moraga. 

Andrew Tawfeek, University of Washington
A Tropical Framework for Using Porteous' Formula
Given a tropical cycle X, one can talk about a notion of "tropical" vector bundles on X having tropical fibers. By restricting our attention to bounded rational sections of these bundles, one can develop a good notion of characteristic classes that behave as expected classically. We present further results on these characteristic classes and use these properties to prove a Porteous' formula for these bundles, which gives a determinantal expression of the fundamental class of degeneracy loci of a (tropical) bundle morphism in terms of their Chern classes. This is a work in progress.

Sebastian Torres, University of Miami
The BGMN conjecture via stable pairs
We construct a semi-orthogonal decomposition on the moduli space of stable rank-two vector bundles on a curve of genus at least two, conjectured by Narasimhan and by Belmans-Galkin-Mukhopadhyay. This is joint work with Jenia Tevelev.

Yi Wei, University of Wisconsin-Madison
Green's Conjecture And The Geometric Syzygy Conjecture in Char p
We study the syzygies of canonical curves of genus g>3 over an algebraically closed field F with char(F)=p>0. With a deformation argument on the moduli space of K3 surfaces, we prove the generic Green's conjecture and a special case of the generic Geometric Syzygy Conjecture, under an assumption of a lower bound on p.

Fei Xiang, UCI
On the Crepant Resolutions and Quiver Moduli Spaces
"It is first observed by McKay that there exists a one-to-one correspondence between the components of the exceptional locus of a resolution and the nontrivial irreducible representations of the group, which is known as McKay Correspondence. The existence of a crepant resolution and the equivalence between the derived categories have been proved in some particular cases.
This poster will be interested in the conjecture that the existence of such a resolution, with some additional restriction, should result in one of the quiver moduli spaces. We will then give the main idea of the recovering process of the resolution from the derived category using Hochschild cohomology. "