COMB in CAMB - Courses

Brill-Noether theory via degeneration
Isabel Vogt (Brown)
Problem Sheets: 1 2

The motivating problem is: what is the geometry of the space of degree d maps from an abstract curve C of genus g to projective space of dimension r? The classic Brill-Noether theorem answers this question when the curve C is general in moduli. In this minicourse, we'll take the perspective that smooth curves are hard, and study this problem by degeneration to well-chosen reducible curves. This approach has the advantage of simplifying the geometric complexity, at the expense of introducing combinatorial complexity.

In the first lecture I'll introduce this topic and a nice degeneration to study the Brill-Noether problem. In the second lecture we'll show that the natural moduli space on the central fiber shows an interesting combinatorial structure. In the third lecture, I'll discuss how to recover global information about general curves from this degeneration, namely the theory of limit linear series and Eisenbud-Harris's regeneration theorem. Finally, in the last lecture, we'll move away from general curves to curves equipped with a (low degree) map to the projective line. In this case, the classic Eisenbud-Harris regeneration theorem does not apply, and I'll discuss how, with my collaborators Eric Larson and Hannah Larson, we solve this problem by mining the deep combinatorial structure on the central fiber related to the affine symmetric group.

Tropical geometry, heights and arithmetic
Farbod Shokrieh (Washington)

The theory of "heights" is essential in studying finiteness questions in diophantine geometry. For example, one of the most spectacular results in number theory is Faltings's theorem (Mordell's conjecture): a curve of genus greater than 1 over the field of rational numbers has only finitely many rational points. A crucial tool in Faltings's proof was a notion of "height of an abelian variety".

In these lectures, we study the interplay between non-archimedean/tropical geometry, arithmetic/Arakelov geometry, and combinatorics/convex geometry arising from the theory of heights of abelian varieties. In particular, we discuss how Faltings's height can be related to invariants arising from tropical geometry and Berkovich analytic spaces. In the case of Jacobians, these invariants relate to combinatorics, convex geometry, and potential theory (electrical network) of metric graphs. If time permits, we will also discuss some applications in the study of analytic invariants on degenerating families of curves and abelian varieties.

Tropical geometry of Bruhat-Tits buildings
Marvin Anas Hahn (Trinity College Dublin)
Problem Sheets: 1

One reason for the interest in groups is that they act on interesting objects. Therefore, it is natural to ask: Given a group, can we construct such an interesting object on which the group acts in a meaningful way? For certain algebraic groups, such constructions give rise to so-called buildings. These buildings are generally (possibly infinite) combinatorial structures that encode group-theoretic information in combinatorial data. In this lecture series, we will focus on "affine buildings" associated with so-called reductive groups. For such groups, there is a close relationship between the corresponding building and tropical geometry. We will show different facets of this rich interaction. In the lecture series we will focus on the group SL_n over a discretely value field, not only because the general construction is quite complicated, but also because for this group the connection to tropical geometry is (so far) the strongest.

The aim of the lecture series is to construct the building of SL_n, to introduce the notion of convexity of this building and to derive applications to the theory of orders and Mustafin varieties. Time permitting, we will also study compactifications of Bruhat-Tits buildings via Berkovich spaces.

Algebraic and tropical Prym varieties
Yoav Len (St Andrews)
Notes
Problem Sheets: 1 2 3

Prym varieties are a class of abelian varieties that show up in the presence of curves with symmetries. Similarly to Jacobians, they classify certain divisors on curves, however, they capture a much larger portion of the moduli space of abelian varieties. They play a key role in rationality questions for threefolds, construction of compact hyper-Kähler manifolds, and the birational geometry of the moduli of abelian varieties. Recent developments in tropical geometry open the way to a combinatorial study of Prym varieties. The course will begin with an overview of the theory of divisors on tropical curves and tropical Jacobians. We will then discuss the intriguing combinatorics of tropical Prym varieties, their relation to algebraic Prym varieties, and potential applications.