impanga 2024/25

I will present the recent developments on free plane curves, mostly focusing on the problem of constructing algebraic surfaces having large Picard numbers and the so-called Numerical Terao's Conjecture. Time permitting, I will deliver some recent progress on Ziegler pairs of line arrangements.

The r-th cactus variety of a subvariety X in a projective space generalises secant variety of X and it is defined using linear spans of finite schemes of degree r. It's original purpose was to study the vanishing sets of catalecticant minors. We propose adding a scheme structure to the cactus variety and we define it via relative linear spans of families of finite schemes over a potentially non-reduced base. In this way we are able to study the vanishing scheme of the catalecticant minors. For X which is a sufficiently large Veronese reembedding of projective variety, we show that r-th cactus scheme and the zero scheme of appropriate catalecticant minors agree on an open and dense subset which is the complement of the (r-1)-st cactus variety/scheme. As an application, we can describe the singular locus of (in particular) secant varieties to high degree Veronese varieties. Based on a joint work with Hanieh Keneshlou.

tba

tba

Stated in 1971, the Zariski multiplicity conjecture asserts that the multiplicity of a reduced complex hypersurface singularity is an embedded topological invariant. More than 50 years later, this problem is still open in general. Several partial answers have been given over the years, with a major breakthrough by J. Fernández de Bobadilla and T. Pełka who recently proved a version of the conjecture for 1-parameter families of isolated singularities. In this talk, I will give an overview of the most significant results that have been obtained.

This is the first of two lectures concerning my joint work with J. F. de Bobadilla, proving Zariski multiplicity conjecture for μ-constant families of isolated hypersurface singularities. In this lecture I will explain the key ingredient of the proof, namely the construction of a representative of monodromy which is an exact symplectomorphism with very good dynamic properties. Given a family of Kähler manifolds over a punctured disk, together with a model with snc central fiber, I will show how to extend it to a symplectic fibration over an annulus, such that over the inner circle ("at radius zero") most choices become irrelevant, in particular the symplectic monodromy can be read off from the dual complex of the central fiber. The resulting fibration, called the A'Campo space, can be applied to many degeneration problems in algebraic geometry. If time permits, I will show how to use it to produce Lagrangian tori approximating those predicted by Mirror Symmetry.

In this introductory lecture, I will describe basics of Heegaard Floer homology (construction and properties) focusing on examples of special importance to experts in singularity theory and low-dimensional algebraic geometry.

In the second lecture, I will explain how to use the A'Campo space, to prove the Zariski conjecture. Just like the classical topological A'Campo construction allows to compute Lefschetz number of monodromy's iterates, our symplectic version yields a spectral sequence converging to its Floer homology. This proof is a slight generalization of the previous work by McLean, in particular it recovers his characterization of multiplicity as the smallest integer m such that the m-th iterate of monodromy has nonvanishing Floer homology. Our more general setting allows us to apply it to each member of a μ-constant family, and infer that the resulting Floer homology - hence multiplicity - stays the same.

Subspace varieties are algebraic varieties whose elements are tensors with bounded multilinear rank. In this talk, we understand how to compute their degrees by computing their volumes. This is joint work with P. Breiding.

Prym theory connects coverings of curves with abelian varieties. The most famous result of the theory, due to Donagi and Smith, is the fact that the Prym map of double coverings of genus 6 curves is 27:1 and is related to 27 lines on a cubic. Apart from that, we know a lot about Prym maps of double coverings and we have some results for cyclic coverings. After a brief introduction to the theory and mentioned results, I will focus on my results about Klein (i.e. Z_2xZ_2) coverings of hyperelliptic curves. The talk is based on joint projects with A. Ortega and with A. Shatsila.