CoAlgGeo HUG - Titles and abstracts

Veronica Arena: The Tautological Ring and Non-tautological Cycles of H*(M_{g,2m})

When studying the cohomology ring of the moduli space of n-pointed curves H^*(M_{g,n}), one can identify a particular subring RH^*(M_{g,n}) called the Tautological ring. This ring has a simple construction via generators, however is very rich and the question of whether or not the equality RH^*=H^* is, in general, hard to answer. Examples of non-tautological classes are provided in work by Graber and Pandharipande and of van Zelm. In a generalisation of the technique used in the latter, in joint work together with S. Canning, E. Clader, R. Haburcak, A.Q. Li, S.C. Mok and C. Tamborini, we construct non-tautological classes for H^*(M_{g,2m}) for infinitely many g,m.

Ana Botero: Volumes of Moduli Spaces, b-Divisors and Conical Zeta Values

What is the volume of the complex universal elliptic curve? There are two ways to think about this question. One is from the point of view of algebraic geometry using intersection theory, another comes from differential geometry using wedge products and differential forms. The compatibility of both approaches is known as Chern-Weil theory. We will see that in the case of the universal elliptic curve, Chern-Weil theory is NOT satisfied, however a satisfactory alternative is provided by the theory of b-divisors. Moreover, the difference of the algebraic- and differential-geometric computation is given by a conical zeta value. This talk is going to be a journey starting from Chern-Weil theory, passing throuh b-divisors then conical zeta values and back. This is work in progress with Claudia Alfes Neumann and Annika Burmester.

Juliette Bruce: Syzygies Beyond Projective Space

A classical idea originating with Hilbert and continuing through the modern work of Lazarsfeld, Voisin, and many others is that homological algebra over the polynomial ring provides many powerful tools for studying the geometry of varieties in projective space. Analogously, one wishes for similar homological tools to study the geometry of varieties embedded in spaces other than projective space. I will discuss my recent work developing such analogous results for sub-varieties of toric varieties via homological algebra over multigraded polynomial rings.

Francesca Carocci: Correlated Gromov-Witten Invariants

In this talk we introduce a geometric refinement of Gromov-Witten invariants for P1-bundles relative to the natural fiberwise boundary structure. We call these refined invariants correlated Gromov-Witten invariants. We will introduce the correlated invariants, discuss their properties and provide some computations in the case of P1-bundles over an elliptic curve. Such invariants are expected to play a role in the degeneration formula for reduced Gromov-Witten invariants for abelian and K3 surfaces. This is a joint work with Thomas Blomme.

Angelica Cueto: Tritangent Planes to Space Sextic Curves: a Tropical Viewpoint

A classical result due to Clebsch from the mid-nineteenth century confirms that every complex space sextic curve (given as an intersection of a quadric and a cubic surface in projective 3-space) has exactly 120 tritangent planes. In this talk we will show how to use combinatorial methods arising from tropical geometry to revisit this classical problem and perform the analogous count over the reals and extensions thereof. This is joint work in progress with Yoav Len, Hannah Markwig and Yue Ren.

Ana-Maria Castravet: Hilbert’s 14’th Problem, Gale Duality and Moduli Spaces

Mukai's counterexamples to Hilbert's 14'th problem rely on the key observation, due to Nagata, that invariant rings for certain vector groups can be identified with Cox rings of certain blow-ups of projective spaces. The Cox ring of a variety encodes a lot of information about the birational geometry of the variety and one can use geometry to answer questions about the finite generation of the Nagata invariant rings. After an introduction to these questions, we will discuss the birational geometry of blow-ups of projective spaces at points in general position. For that, we will explore Gale duality, a correspondence between sets of n=r+s+2 points in projective spaces P^r and P^s. For small values of s, this duality has a remarkable geometric manifestation: the blow-up of P^r at n points can be realized as a moduli space of vector bundles on the blow-up of P^s at the Gale dual points.

Samantha Fairchild: Detecting Algebraic Properties in Geometric Objects

Though the uniformization theorem guarantees an equivalence of Riemann surfaces and smooth algebraic curves, moving between analytic and algebraic representations is inherently transcendental. We will build a family of non-hyperelliptic surfaces of genus at least 3 where we present the Riemann surface through the Schottky uniformization (identifying pairs of circles via Möbius transformations in the complex plane). We will focus on the example of dihedral symmetry, giving properties of the canonical embedding, a real structure with the maximal number of connected components (an M-curve), and degenerations to the boundary of moduli space.

Alheydis Geiger: Real Tropical Quartic Curves and their Bitangents

The question of lifting tropical varieties in a certain relation (e.g. tangents to a curve) to algebraic varieties satisfying the same relation is of particular interest because of the superabundance phenomenon in tropical geometry. For certain tropical quartic curves the existing techniques can not predict the lifting behavior of its bitangents over the real numbers. We close this gap by using patchworking techniques. Further, we provide an analysis of the combinatorial types of real tropical quartic curves according to their real topology and number of real bitangents. This talk will start with an introduction to tropical geometry and revisit the relevant results of lifting tropical bitangents from works by Cueto and Markwig, and from joined work with Marta Panizzut.

Victoria Hoskins: Geometric Invariant Theory for Non-reductive Groups

I will start with a brief overview of Mumford's GIT for reductive groups and illustrate what can go wrong for non-reductive groups with some examples. I will survey progress on extending this to non-reductive groups due to Berczi and Kirwan, then explain how to generalise their results to affine and relative settings, and if there is time, outline how this can be applied to construct moduli of representations of quivers over truncated polynomial rings (based on joint works with E. Hamilton, J. Jackson and T. Vernet).

Milena Hering: Stability of Toric Vector Bundles

I will give an introduction to toric vector bundles, with a view towards their stability. I will explain applications to tangent bundles, in particular we show that there exists a polarisation such that the tangent bundle is stable on a toric surface if and only if the surface is a blow up of the projective plane. We also use toric techniques to produce a counterexample to a question of Ein, Lazarsfeld and Mustopa regarding the stability of syzygy bundles. This is joint work with Lucie Devey, Katharina Jochemko, Benjamin Nill, and Hendrik Süss.

Sabrina Pauli: A Tropical Correspondence Theorem for Quadratic Plane Curve Counts

The number of rational plane curves of a given degree satisfying point conditions is a classical enumerative invariant. In the real setting, restoring invariance requires a signed count, while more generally, curves should be counted with a weight in the Grothendieck-Witt ring of quadratic forms.

To compute such counts, one can translate the problem into tropical geometry, where it reduces to a weighted count of certain graphs with point conditions. This significantly simplifies the computation. In this talk, I will present a tropical correspondence theorem for quadratic plane curve counts.

Ragni Piene: Toric Interpolation

A classical interpolation problem in algebraic geometry is to find a curve of a given type that passes through, or interpolates, a given set of points. A classical interpolation problem in differential geometry is to consider the normal planes to a curve in 3-space and look for a curve which has these planes as osculating planes, i.e., to find the evolute of the given curve. The toric interpolation problem is the following: Given a toric variety X, does there exist a toric variety Y containing X such that the tangent space to Y at any point of X is equal to the osculating space to X at that point? With Alicia Dickenstein and Sandra Di Rocco we proved that Y – the toric interpolant – exists and is unique.

In this talk I will give a gentle introduction to toric varieties, defined by lattice polytopes, and their osculating spaces, and explain the construction of the toric interpolant. As illustrations, several examples will be given.

Milena Wrobel: A Combinatorial Description of the Gorenstein Index for Varieties with Torus Action

Toric Fano varieties build an important example class in algebraic geometry as their high symmetry allows to describe them in a purely combinatorial manner via the so-called Fano polytopes. The anticanonical complex is a combinatorial tool that was invented to extend the features of the Fano polytope to wider classes of varieties. In this talk we give an outline of the construction of anticanonical complexes and show how they can be used to read off the Gorenstein index of Fano varieties with torus action of higher complexity in full analogy to the toric Fano polytope.

Claudia Yun: The Topology of the Moduli Spaces of Tropical Unramified p-Covers

Tropical geometry is a technique that allows us to study algebro-geometric objects through combinatorics. Let p be a prime number. The moduli space M_{g,Z/p} parametrizes isomorphism classes of étale (Z/p)-Galois covers of smooth curves of genus g. It is itself an étale covering of M_g, the moduli space of smooth curves of genus g.

On the other hand, an abstract tropical curve is a decorated metric graph. We study an associated moduli space of unramified tropical (Z/p)-covers of curves of genus g. Its rational homology is conjecturally identified with the top weight cohomology of M_{g,Z/p}. When g > 2, we identify a nested sequence of contractible loci in the moduli space of tropical covers and show that it is simply connected. In the case g=2, we also determine the homotopy type of the tropical space, showing that it is contractible for small p and is a wedge of spheres for p > 3. This is joint work with Yassine El Maazouz, Paul Alexander Helminck, Felix Röhrle, and Pedro Souza.

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