Related papers: Curves of maximal moduli on K3 surfaces

We describe Baily-Borel, toroidal, and geometric -- using the KSBA stable pairs -- compactifications of some moduli spaces of K3 surfaces with a nonsymplectic automorphism of order $3$ and $4$ for which the fixed locus of the automorphism…

We study genus 3 hyperelliptic curves which have an extra involution. The locus $\L_3$ of these curves is a 3-dimensional subvariety in the genus 3 hyperelliptic moduli $\H_3$. We find a birational parametrization of this locus by affine…

A $K3$ surface with an ample divisor of self-intersection 2 is a double cover of the plane branched over a sextic curve. We conjecture that a similar statement holds for the generic couple $(X,H)$ with $X$ a deformation of $(K3)^{[n]}$ and…

We prove that for any of a wide class of elliptic surfaces $X$ defined over a number field $k$, if there is an algebraic point on $X$ that lies on only finitely many rational curves, then there is an algebraic point on $X$ that lies on no…

The Green-Griffiths-Lang conjecture stipulates that for every projective variety $X$ of general type over ${\mathbb C}$, there exists a proper algebraic subvariety of $X$ containing all non constant entire curves $f:{\mathbb C}\to X$. Using…

Given a smooth, irreducible, projective surface $S$, let $g(S)$ be the minimum geometric genus of an irreducible curve that moves in a linear system of positive dimension on $S$. We determine the value of this birational invariant for a…

Given d in IN, we prove that all smooth K3 surfaces (over any field of characteristic p other than 2,3) of degree greater than 84d^2 contain at most 24 rational curves of degree at most d. In the exceptional characteristics, the same bounds…

Let $\V_{d,n}$ be the Severi variety of irreducible plane curves of degree $d\ge 4$ having $n$ nodes, with $0\le n \le \binom{d-1}{2}-1$. We prove that for every $[\ol C]\in \V_{d,n}$, the infinitesimal variation of the Hodge structure of…

Let X be an Abelian surface and C a holomorphic curve in X representing a primitive homology class. The space of genus g curves in the class of C is g dimensional. We count the number of such curves that pass through g generic points and we…

In "Curves on Heisenberg invariant quartic surfaces in projective 3-space", Eklund showed that a general $(\mathbb{Z}/2\mathbb{Z})^{4}$-invariant quartic K3 surface contains at least $320$ conics. In this paper we analyse the field of…

We show that every possible value for the Clifford index and gonality of a curve of a given genus on a $K3$ surface occurs.

We present three families of pairs of geometrically non-isomorphic curves whose Jacobians are isomorphic to one another as unpolarized abelian varieties. Each family is parametrized by an open subset of P^1. The first family consists of…

Let k be an algebraically closed field of characteristic p. Let X(p^e;N) be the curve parameterizing elliptic curves with full level N structure (where p does not divide N) and full level p^e Igusa structure. By modular curve, we mean a…

Let $X$ be a geometrically irreducible smooth projective curve, of genus at least three, defined over the field of real numbers. Let $G$ be a connected reductive affine algebraic group, defined over $\mathbb R$, such that $G$ is nonabelian…

For every $q=l^3$ with $l$ a prime power greater than 2, the GK curve $X$ is an $F_{q^2}$-maximal curve that is not $F_{q^2}$-covered by any $F_{q^2}$-maximal Deligne-Lusztig curve. Interestingly, $X$ has a very large $F_{q^2}$-automorphism…

Let $(S,H)$ be a general primitively polarized $K3$ surface. We prove the existence of curves in $|\mathcal O_S(nH)|$ with $A_k$-singularities and corresponding to regular points of the equisingular deformation locus. Our result is optimal…

In this paper we classify curves of genus two over a perfect field k of characteristic two. We find rational models of curves with a given arithmetic structure for the ramification divisor and we give necessary and sufficient conditions for…

We construct the moduli of twisted sheaves on a projective variety. Then we generalize known results on the moduli space of usual sheaves on a K3 surface to the twisted case. Thus we consider the non-emptyness, the deformation type and the…

We introduce a notion of good cohomology for multiple lines in $\mathbb{P}^3$ and we classify multiple lines with good cohomology up to multiplicity 4. In particular, we show that the family of space curves of degree d, not lying on a…

We prove K-stability for infinitely many smooth members of the family 2.19 of the Mukai-Mori classification.