Seminar
Programme 2024 – 2025
A story of degenerations in enumerative geometry
The basic principle of degeneration has been present in enumerative geometry since the 19th century. With Gromov-Witten theory in the 1990s, the idea grew into a robust set of techniques. I'll outline the basic ideas and highlight connections to tropical geometry, the double ramification cycle, and the cohomological study of the moduli space of curves.
Geometry of 3-dimensional del Pezzo fibrations in positive characteristic
A 3-dimensional del Pezzo fibration \(X \to C\) is one of the possible outcomes of the MMP. In positive characteristic, the generic fibre is defined over an imperfect field, where regularity is weaker than smoothness. We extend classical results due to Enriques, Manin, and Iskovskikh.
Genus 4 curves: invariants and reconstruction
We introduce algebraic invariants for non-hyperelliptic genus 4 curves, characterising isomorphism classes over an algebraically closed field of characteristic zero, along with a reconstruction algorithm generalising that of Mestre.
Brunn-Minkowski in \(SO_n(\mathbb{R})\)
We prove a conjecture by Breuillard and Green predicting a lower bound for doubling in \(SO_n(\mathbb{R})\), establishing a Brunn-Minkowski type inequality and a stability result.
Refined invariants for real curves
New invariants of the projective plane arising from signed enumeration of real algebraic curves of genus 1 and 2, admitting a refinement similar to Mikhalkin's genus zero case. Two different rules of signs surprisingly lead to the same invariants. Joint work with Eugenii Shustin.
On a problem of Erdős about sums of distinct powers of integers
Joint work with Maximilian F. Hasler. A new lower estimate of the counting function of positive integers that are sums of distinct powers of 3 and 4, a problem raised by Erdős in 1996.
No seminar — ColboisFest
On the dynamics of endomorphisms of affine surfaces
For polynomial maps of \(K^2\), Favre and Jonsson (2007) showed the dynamical degree is an algebraic integer of degree ≤ 2. We generalise this to any affine surface by studying the dynamics on a space of valuations associated to \(S\).
Correlated Gromov-Witten invariants (joint with Francesca Carocci)
A geometric refinement of Gromov-Witten invariants for \(\mathbb{P}^1\)-bundles relative to the fiberwise boundary, with computations over elliptic curves. Expected to play a role in degeneration formulas for abelian and K3 surfaces.
The modular support problem
Inspired by an Erdős problem and the analogy between roots of unity and singular moduli, Francesco Campagna and I studied the support problem with Hilbert class polynomials \(H_D(T)\) instead of \(T^n - 1\).
Subrank of bilinear maps
The subrank of \(f: U \times V \to W\) measures independent scalar multiplications using \(f\) once. We discuss recent work determining the subrank of any sufficiently general \(f\) and asymptotics of generic border subrank.
Tropical combinatorics of the 2D Toda lattice
The tropical approach of Kodama-Williams for the KP hierarchy generalises to the 2D Toda lattice, answering a question of Kodama. Joint work in progress with Betancourt, Posch and Reda.
On the \(K(\pi,1)\)-problem for abelian arrangements
Recent advances on the \(K(\pi,1)\)-problem for arrangements in tori, products of elliptic curves, and connected Abelian Lie groups. Joint work with Bibby, D'Alì, Girard, Paolini, Riedel. Note: talk at 4pm, room E213.
No seminar — Conference Géométrie des groupes et groupes de Cremona
On the rationality of real conic bundles
We review rationality techniques over \(\mathbb{R}\) and provide new positive and negative results for real conic bundles. Joint work with Alena Pirutka.
Classification of non-F-split del Pezzo surfaces
Non-F-split del Pezzo surfaces exist only in degrees 3, 2, and 1. We give a geometric characterisation in degree 1. Joint work with Réka Wagener.
Abstract to follow.