Related papers: The flex divisor of a K3 surface

This thesis is devoted to the study of abelian automorphism groups of surfaces and $3$-folds of general type over complex number field $\Bbb C$. We obtain a linear bound in $K^3$ for abelian automorphism groups of $3$-folds of general type…

In this paper, we study the relative anti-canonical divisor $-K_{X/Y}$ of an algebraic fiber space $\phi: X \to Y$, and we reveal relations among positivity conditions of $-K_{X/Y}$, certain flatness of direct image sheaves, and variants of…

The flex locus parameterizes plane cubics with three collinear cocritical points under a projection, and the gothic locus arises from quadratic differentials with zeros at a fiber of the projection and with poles at the cocritical points.…

In this paper we show that every complex hypersurface $A$ in a Stein manifold $X$ with $H^2(X;\mathbb Z)=0$ is the divisor of a holomorphic function $f$ on $X$ whose critical points are precisely the singular points of $A$. A similar result…

We consider threefolds that admit a fibration by K3 surfaces over a nonsingular curve, equipped with a divisorial sheaf that defines a polarisation of degree two on the general fibre. Under certain assumptions on the threefold we show that…

Let $X$ be a smooth projective surface over the complex number field and let $L$ be a nef-big divisor on $X$. Here we consider the following conjecture; If the Kodaira dimension $\kappa(X)\geq 0$, then $K_{X}L\geq 2q(X)-4$, where $q(X)$ is…

We give a formula in terms of multidimensional resultants for an equation for the flex locus of a projective hypersurface, generalizing a classical result of Salmon for surfaces. Using this formula, we compute the dimension of this flex…

We give many examples in which there exist infinitely many divisorial conditions on the moduli space of polarized K3 surfaces $(S,H)$ of degree $H^2=2g-2$, $g \geq 3$, and Picard number $rk N(S)=\rho(S)=2$ such that for a general K3 surface…

Let X be a K3 surface with an involution g which has non-empty fixed locus X^g and acts non-trivially on a non-zero holomorphic 2-form. We shall construct all such pairs (X, g) in a canonical way, from some better known double coverings of…

Let $X$ be the variety of flexes of plane cubics. We prove that (1) $X$ is an irreducible rational algebraic variety endowed with a faithful algebraic action of ${\rm PSL}_3$; (2) $X$ is ${\rm PSL}_3$-equivariantly birationally isomorphic…

Let $D$ be a reduced divisor in $\mathbb P^n_k$ for an algebraically closed field $k$ of positive characteristic $p > 0$. We prove that if $(\mathbb P^n_k, D)$ is Frobenius liftable modulo $p^2$, then $D$ is a toric divisor. As a corollary,…

Given two vector bundles E and F on a variety X and a morphism from Sym^2(E) to F, we compute the cohomology class of the locus in X where the kernel of this morphism contains a quadric of prescribed rank. Our formulas have many…

Let $X/ \mathbb{C}$ be a K3 surface with complex multiplication by the ring of integers of a CM field $E$. We show that $X$ can always be defined over an Abelian extension $K/E$ explicitly determined by the discriminant form of the lattice…

Let $(S,D)$ be a minimal log pair of general type with $S$ a smooth projective surface and $D$ a simple normal corssing reduced divisor on $S$. We assume that its log canonial linear system $|K_S+D|$ is composed of a penciel, let $f\colon…

For a projective variety X, a line bundle L on X and r a natural number we consider the r-th Brill-Noether locus W^r(L,X):={\eta\in Pic^0(X)|h^0(L+\eta)\geq r+1}: we describe its natural scheme structure and compute the Zariski tangent…

In this paper we prove a conjecture about the dimension of linear systems of surfaces of degree d in P^3 through at most eight multiple points in general position.

The divisor graph is the non oriented graph whose vertices are the positive integers, and edges are the {a,b} such that a divides b or b divides a. Let F(x,y) be the maximum number of integers<= x belonging in one of y pairwise disjoint…

In this paper, we study the Severi variety $V_{L,g}$ of genus $g$ curves in $|L|$ on a general polarized K3 surface $(X,L)$. We show that the closure of every component of $V_{L,g}$ contains a component of $V_{L,g-1}$. As a consequence, we…

We extend results on generic strange duality for K3 surfaces by showing that the proposed isomorphism holds over an entire Noether-Lefschetz divisor in the moduli space of quasipolarized K3s. We interpret the statement globally as an…

Totally invariant divisors of endomorphisms of the projective space are expected to be always unions of linear spaces. Using logarithmic differentials we establish a lower bound for the degree of the non-normal locus of a totally invariant…