Algebraic Geometry / Moduli Seminar 

We compute the Kodaira dimension and the Iitaka fibration of the universal Picard variety parameterizing line bundles of degree $d$ on curves of genus $g$ under the (technical) assumption that $\gcd (d-g+\mathrm{1,2}g-2)=1$. We also give partial results for arbitrary degrees $d$ and we investigate for which degrees the universal Picard varieties are birational. This is a joint work with G. Bini and C. Fontanari.

Gromov-Witten invariants count the number of pseudo-holomorphic curves. One often writes them in terms of generating functions. Occasionally, it posses some very beautiful properties such as being a quasi-modular form. In the talk, we will explain this phenomenon for elliptic orbifold ${\mathbb{P}}^1$. This is a joint work with Milanov, Krawitz and Shen.

The involutions of K3 surfaces have been completely classified by Nikulin. Can we extend his results to holomorphic symplectic manifolds, the natural generalization of K3 surfaces in higher dimension? I will describe some (very partial) results obtained recently in this direction.

I will describe a classical problem going back to 1848 (Steiner, Cayley, Salmon,...) and a solution using simple techniques, but techniques that one would never really have thought of without ideas coming from string theory (Gromov-Witten invariants, BPS states) and modern geometry (the Maulik-Nekrasov-Okounkov-Pandharipande conjecture). In generic families of curves $C$ on a complex surface $S$, nodal curves -– those with the simplest possible singularities - appear in codimension 1. More generally those with $d$ nodes occur in codimension $d$. In particular a d-dimensional linear family of curves should contain a finite number of such $d-$nodal curves. The classical problem - at least in the case of $S$ being the projective plane - is to determine this number. The Göttsche conjecture states that the answer should be topological, given by a universal degree $d$ polynomial in the four numbers $C.C,c_1(S).C,c_1(S{)}^2$ and $c_2(S)$. There are now proofs in various settings; a completely algebraic proof was found recently by Tzeng. I will explain a simpler approach which is joint work with Martijn Kool and Vivek Shende.

In a paper of '94 C. Simpson constructed moduli spaces for a large class of objects, namely semistable $\Lambda$-modules. Main examples of this include flat connections, Higgs bundles, connection along foliations and logaritmic connections. We shall show that Simpson's axioms for the algebra $\Lambda$ canonically associate to it a Lie algebroid and an element of the second cohomology of the algebroid, so that the objects that one get through Simpson's formalism are Lie algebroid connections whose curvature is controlled by the second cohomology class.