Related papers: The enumerative geometry of K3 surfaces and modula…

It is one of the wonderful ``coincidences'' of the theory of finite groups that the simple group G of order 25920 arises as both a symplectic group in characteristic 3 and a unitary group in characteristic 2. These two realizations of G…

A key result for syzygies of curves is Voisin's proof of Green's conjecture for the canonical embedding of a general curve of any genus. Her primary tools were the Lazarsfeld Mukai bundle on a K3 surface and a representation of Koszul…

By a theorem of Wahl, for canonically embedded curves which are hyperplane sections of K3 surfaces, the first gaussian map is not surjective. In this paper we prove that if C is a general hyperplane section of high genus (greater than 280)…

For a line bundle $L$ on a smooth projective surface $X$ and nonnegative integers $k_1, \ldots, k_N$, Okounkov \cite{Oko} introduced the reduced generating series $\big \langle {\rm ch}_{k_1}^{L} \cdots {\rm ch}_{k_N}^{L} \big \rangle'$ for…

A conjecture expressing genus 1 Gromov-Witten invariants in mirror-theoretic terms of semi-simple Frobenius structures and complex oscillating integrals is formulated. The proof of the conjecture is given for torus-equivariant Gromov -…

For a line bundle $L$ on a smooth projective surface $X$ and nonnegative integers $k_1, \ldots, k_N$, Okounkov \cite{Oko} introduced the reduced generating series $\big \langle {\rm ch}_{k_1}^{L} \cdots {\rm ch}_{k_N}^{L} \big \rangle'$ for…

This article surveys some recent work of the author on Hilbert modular fourfolds X. After some preliminaries on the cohomology and special, codimension 2 cycles Z on X of Hirzebruch-Zagier type, a proof of the Tate conjecture for X over…

Let $f:X\to C$ be a family of semistable K3 surfaces with non-empty set $S$ of singular fibres having infinite local monodromy. Then, when the so called Arakelov-Yau inequality reaches equality, we prove that $C\setminus S$ is a modular…

Continuing our work in \cite{1805.04894}, this article is devoted to proving that open-closed Gromov-Witten invariants of $K_{\mathbb{P}^2/\mu_3}$ are quasi-meromorphic modular forms, and generating functions of open Gromov-Witten…

Using obstruction bundles, composition law and the localization formula, we compute certain 3-point genus-0 Gromov-Witten invariants of the Hilbert scheme of 3-points on the complex projective plane. Our results partially verify Ruan's…

Let C be a Brill-Noether-Petri curve of genus g\geq 12. We prove that C lies on a polarized K3 surface, or on a limit thereof, if and only if the Gauss-Wahl map for C is not surjective. The proof is obtained by studying the validity of two…

This article is an expanded version of talks given by the authors in Oberwolfach, Bochum, and at the Fano Conference in Torino. Some new results (e. g. the material concerning flag varieties, Quot spaces over $\P^1$, and the generalized…

Based on the result on derived categories on K3 surfaces due to Mukai and Orlov and the result concerning almost-prime numbers due to Iwaniec, we remark the following fact: For any given positive integer N, there are N (mutually…

We provide an inductive algorithm computing Gromov-Witten invariants in all genera with arbitrary insertions of all smooth complete intersections in projective space. We also prove that all Gromov-Witten classes of all smooth complete…

We prove that the gonality among the smooth curves in a complete linear system on a $K3$ surface is constant except for the Donagi-Morrison example. This was proved by Ciliberto and Pareschi under the additional condition that the linear…

We generalize the formula of Gromov-Witten-Welschinger invariants of $\mathbb{C} P^3$ established by E. Brugall\'e and P. Georgieva in [BG16b] to $\mathbb{C} P^3\#\overline{\mathbb{C} P}^3$. Using pencils of quadrics, some real and complex…

We study the geometry of varieties parametrizing degree d rational and elliptic curves in P^n intersecting fixed general linear spaces and tangent to a fixed hyperplane H with fixed multiplicities along fixed general linear subspaces of H.…

The Prym-Green Conjecture predicts that the resolution of a generic n-torsion paracanonical curve of every genus is natural. The conjecture has mostly been studied so far for level 2, that is, for Prym-canonical curves. Using a construction…

In the first part of the paper, we give an explicit algorithm to compute the (genus zero) Gromov-Witten invariants of blow-ups of an arbitrary convex projective variety in some points if one knows the Gromov-Witten invariants of the…

We show that if over some number field there exists a certain diagonal plane cubic curve that is locally solvable everywhere, but that does not have points over any cubic galois extension of the number field, then the algebraic part of the…