Starting from a symmetric group S_d, we introduce a topological space Hur(S_d) parameterising finite configurations of points in the complex plane with the extra information of an S_d-valued monodromy. Notably, an application of Riemann-Roch ensures that the moduli spaces of curves with directed punctures tend to arise, up to homotopy, as connected components of Hur(S_d). The stable homology of connected components of Hur(S_d) agrees with the homology of the double loop space of a certain space X_d. One can explicitly compute the rational cohomology of X_d, and in a range increasing with d the rational cohomology of its double loop space. Letting d go to infty, one recovers the Mumford conjecture on stable rational cohomology of moduli of curves, originally proved by Madsen-Weiss. I will sketch the argument, and I will report on a possible strategy to obtain information about the unstable cohomology of moduli of curves based on the higher syzygies of the cohomology of X_d.


 I will talk about some ongoing collaborative work with David Klompenhouwer and Paolo Rossi. Starting from the question “what is the relationship between the DR hierarchies and the analogous constructions made with admissible covers”, we realized that in fact in Hurwitz-land even three-pointed invariants can be encoded in a way to give the generating functions good structural properties similar to the DR case. Further, a large class of invariants can be reconstructed from WDVV and minimal initial condition. In this talk we focus on the compact type/one part case (one branch point corresponds to full ramification) where the structure is more transparent.


An extension of a curve C in P^N is a surface S in P^{N+1} such that C is a hyperplane section of S (or, more generally, an r-dimensional variety Y in P^{N+r−1} such that C is a linear section of Y). There is an infinitesimal theory of extensions which is akin to the infinitesimal theory of deformations. The dimension of the space of infinitesimal extensions of C is the corank of a certain Gaussian map associated to the polarized curve C. Under certain regularity conditions, if the infinitesimal extension theory of C is unobstructed then there exists an extension of C of large dimension which packages together all surface extensions of C.

I will present some classification results for extensions of (special) k-gonal and plane curves, and describe the relevant cokernels of Gaussian maps, after joint work with Ciro Ciliberto.


We develop a novel approach to the Brill–Noether theory of curves endowed with a degree k cover of the projective line, via Bridgeland stability conditions on elliptic K3 surfaces.

We first develop the Brill–Noether theory on elliptic K3 surfaces via the notion of Bridgeland stability type for objects in their derived category. As a main application, we show that curves on elliptic K3 surfaces serve as the first known examples of smooth k-gonal curves which are general from the viewpoint of Hurwitz–Brill–Noether theory. In particular, we provide new proofs of the main non-existence and existence results in Hurwitz–Brill–Noether theory. Finally, we construct explicit examples of curves defined over number fields which are general from the perspective of Hurwitz–Brill–Noether theory. Joint work with Soheyla Feyzbakhsh and Andres Rojas.


We will report about the joint work [C. Ciliberto, C. Galati, On nodal deformations of singular surfaces in P3, arXiv:2602.09177], studying nodal deformations of singular surfaces S in the complex projective space. We will in particular consider the case $S$ has ordinary isolated singularities and the case where S is double along a line.


In this talk I will define a Brauer-Severi variety Q->B which is universal in the sense that every other one with related invariants is a pullback of Q. The geometry of these varieties B will be discussed as well as some applications to period-index problems. This is joint work with Daniel Huybrechts.


We investigate the scrollar invariants of the normalization of a general curve of given geometric genus lying in the linear system of a polarized toric surface corresponding to an h-transverse polygon. Through the well-known reduction from algebro-geometric problems to tropical problems, this is obtained by computing the rank of particular divisors on a certain type of floor decomposed tropical curve, and showing the realizability of such tropical curves. This talk is based on joint work with Karl Christ and Ilya Tyomkin.


Brill-Noether theory is the study of line bundles on algebraic curves.  A series of results in the 80's describe the Brill-Noether theory of a general element of the moduli space of curves.  More recently, many researchers have become interested in the Brill-Noether theory of a general element of the Hurwitz space.  We will begin with a gentle introduction to this field of study, and then survey some of the recent results in this area.  We will conclude with recent results and even more recent conjectures on etale double covers of curves, a subject known as Prym-Brill-Noether theory.


We will describe results with Ishan Levy proving that the homology of Hurwitz spaces stabilizes. In simple cases, we compute the stable value of this homology. In general, the stable homology seems difficult to compute, but we can still show it agrees with the homology of simpler Hurwitz spaces. As one application of these results, we prove an asymptotic version of the Picard rank conjecture, which is a conjecture predicting the rational Picard group of simply branched Hurwitz spaces.


I will report on ongoing work wit C. Ciliberto and S. Torelli on various questions concerning Hurwitz spaces with prescribed ramification. These spaces parametrize degree-d covers between curves of fixed genera with prescribed ramification at marked points on the domain.  


 Brill--Noether theory studies the maps of curves C to projective spaces. The classical Brill--Noether theorem (established by work of Eisenbud, Fulton, Geiseker, Griffiths, Harris, Lazarsfeld) describes the geometry of this space of maps when C is a general curve. However, the theorem fails for special curves, notably curves that are already equipped with some unexpected map to a projective space. The first case of this is when C is a low-degree cover of the projective line. For general such covers, the Hurwitz--Brill--Noether theorem (joint with E. Larson and I. Vogt) provides a suitable analogue. After reviewing this work, I'll present new results (joint with S. Vemulapalli) regarding the next natural case: when C is equipped with an embedding in the projective plane. Our theorem for plane curves is a special case of a more general result for smooth curves on Hirzebruch surfaces.


In continuation of Aaron Landesman's talk, I will outline our proof of homological stability for Hurwitz spaces.



The double ramification (DR) cycle parametrizes genus g curves admitting maps to the projective line with specified ramification profiles over two points. The DR cycle depends polynomially in the parts of the ramification profiles, so we may consider its top degree part, a homogeneous polynomial of degree 2g with coefficients in the Chow ring of the moduli space of stable curves. Conjecturally, this top degree part also describes the asymptotic growth of the corresponding admissible covers cycle. After discussing this conjecture, I will switch to a different perspective on the top degree part of the DR cycle, presenting joint work with Y. Bae and S. Molcho coming from the intersection theory of compactified Jacobians.


In my talk, I will discuss a tropical approach to classical irreducibility problems in Algebraic Geometry. I will explain how to prove the irreducibility of Moduli spaces of curves, Severi varieties, and Hurwitz schemes in arbitrary characteristic by investigating the properties of tropicalization of families of curves. The talk is based on a series of joint works with Karl Christ and Xiang He.


Generalized Fermat manifolds (GFM) are defined by a system of Fermat-type equations and extend the classical Fermat manifolds, which are given by a single equation.

The aim of this talk is to study certain quotients of these manifolds. In particular, we focus on GFMs that are Calabi–Yau manifolds and investigate specific automorphisms of finite order acting on them.

We show that the resulting quotients are either singular Calabi–Yau manifolds or Log Enriques varieties of index 2. Log Enriques varieties have been introduced only recently by several authors; they can be viewed as singular analogues of Enriques manifolds, which themselves generalize Enriques surfaces. Moreover, we prove that, with only a few exceptions, these quotients have terminal singularities. The results are contained in a joint work in progress with A. Palomino.


 The talk will focus on determining the irreducible components of Severi varieties of primitive class curves on general (1,d)-polarized abelian surfaces. These components are completely determined by the maximal factorization of the maps from the normalized curves to the abelian surface. The proof uses the degeneration originally employed by Bryan and Leung in the computation of Severi degrees. Time permitting, I will also briefly discuss another application involving singular primitive class curves on K3 or abelian surfaces, inspired by the SHGH conjecture, which can be approached by similar methods.


We compute the stable rational cohomology of the Hurwitz space parametrizing simply branched degree 3 covers. This result was already known by Landesman and Levy, who proved the stabilization result for all Hurwitz spaces parametrizing simply branched degree d covers. 

However, we give an alternative proof and obtain an explicit stable range, combining Gorinov-Vassiliev's method and point counting over finite fields.

This is work in progress with Jonas Bergström.