Related papers: A Simple Proof that Rational Curves on K3 are Noda…

We consider a general primitively polarized K3 surface $(S,H)$ of genus $g+1$ and a 1-nodal curve $\widetilde C\in |H|$. We prove that the normalization $C$ of $\widetilde C$ has surjective Wahl map provided $g=40,42$ or $\ge 44$.

Given a general polarized $K3$ surface $S\subset \mathbb P^g$ of genus $g\le 14$, we study projections $S\hookrightarrow \mathbb P^g\dashrightarrow \mathbb P^2$ of minimal degree and their variational structure. In particular, we prove that…

We study the projective models of complex K3 surfaces polarized by a line bundle L such that all smooth curves in |L| have non-general Clifford index. Such models are in a natural way contained in rational normal scrolls. We use this study…

We prove that the spaces of rational curves on del Pezzo surfaces are either irreducible or empty, with a unique exception.

The surfaces considered are real, rational and have a unique smooth real $(-2)$-curve. Their canonical class $K$ is strictly negative on any other irreducible curve in the surface and $K^2>0$. For surfaces satisfying these assumptions, we…

We construct geometric compactifications of the moduli space $F_{2d}$ of polarized K3 surfaces, in any degree $2d$. Our construction is via KSBA theory, by considering canonical choices of divisor $R\in |nL|$ on each polarized K3 surface…

We give a practical formula for counting irreducible nodal genus-three plane curves that a fixed generic complex structure on the normalization. As an intermediate step, we enumerate rational plane curves that have a $(3,4)$-cusp.

We determine all possible configurations of rational double points on complex normal algebraic K3 surfaces, and on normal supersingular K3 surfaces in characteristic p > 19.

A rational triangle is a triangle with rational sides and rational area. A Heron triangle is a triangle with integral sides and integral area. In this article we will show that there exist infinitely many rational parametrizations, in terms…

We prove that any smooth rational projective surface over the field of complex numbers has an open covering consisting of 3 subsets isomorphic to affine planes.

This is a survey of the geometry of complex cubic fourfolds with a view toward rationality questions. Topics include classical constructions of rational examples, Hodge structures and special cubic fourfolds, associated K3 surfaces and…

A K3 surface is a quaternionic analogue of an elliptic curve from a view point of moduli of vector bundles. We can prove the algebraicity of certain Hodge cycles and a rigidity of curve of genus eleven and gives two kind of descriptions of…

Let k=F_q be a finite field of even characteristic. We obtain in this paper a complete classification, up to k-isomorphism, of non singular quartic plane curves defined over k. We find explicit rational normal models and we give closed…

The aim of this paper is to describe algebraic K3 surfaces with an even set of rational curves or of nodes. Their minimal possible Picard number is nine. We completely classify these K3 surfaces and after a carefull analysis of the divisors…

We develop a novel approach to the Brill-Noether theory of curves endowed with a degree k cover of the projective line via Bridgeland stability conditions on elliptic K3 surfaces. We first develop the Brill-Noether theory on elliptic K3…

We generalize results by Wakabayashi and Orevkov about rational cuspidal curves on the projective plane to that on $\mathbb{Q}$-homology projective planes. It turns out that the result is exactly the same as the projective plane case under…

We classify real families of minimal degree rational curves that cover an embedded rational surface. A corollary is that if the projective closure of a smooth surface is not biregular isomorphic to the projective closure of the unit-sphere,…

We show several examples of integrable systems related to special K3 and rational surfaces (e.g., an elliptic K3 surface, a K3 surface given by a double covering of the projective plane, a rational elliptic surface, etc.). The construction,…

The aim of this paper is to prove that a K3 surface is the minimal model of the quotient of an Abelian surface by a group $G$ (respectively of a K3 surface by an Abelian group $G$) if and only if a certain lattice is primitively embedded in…

We study qualitative aspects of the Welschinger-like $\mathbb Z$-valued count of real rational curves on primitively polarized real $K3$ surfaces. In particular, we prove that with respect to the degree of the polarization, at logarithmic…