Related papers: On the arithmetic self-intersection number of the …
Using the discriminant modular form and the Noether formula it is possible to write the admissible self-intersection of the relative dualising sheaf of a semistable hyperelliptic curve over a number field or function field as a sum, over…
We classify completely the intersections of the Hermitian curve with parabolas in the affine plane. To obtain our results we employ well-known algebraic methods for finite fields and geometric properties of the curve automorphisms. In…
Given a locally free coherent sheaf on a smooth algebraic surface, we consider the Quot-scheme parametrizing zero-dimensional quotients of the sheaf and find the corresponding motivic class in the Grothendieck ring of algebraic varieties.
We analyse infinitesimal deformations of morphisms of locally free sheaves on a smooth projective variety $X$ over an algebraically closed field of characteristic zero. In particular, we describe a differential graded Lie algebra…
We study moduli of semistable twisted sheaves on smooth proper morphisms of algebraic spaces. In the case of a relative curve or surface, we prove results on the structure of these spaces. For curves, they are essentially isomorphic to…
We determine necessary conditions for ample divisors in arbitrary genus as well as for very ample divisors in genus 2 and 3. We also compute the intersection numbers $\lambda^9$ and $\lambda_{g-1}^3$ in genus 4. The latter number is…
In the present paper, we focus on a weighted version of the Bounded Negativity Conjecture which predicts that for every smooth projective surface in characteristic zero the self-intersection numbers of reduced and irreducible curves are…
Let K be a number field, O_K the ring of integers of K and X a stable curve over O_K of genus g >= 2. In this note, we will prove a strict inequality ( (K_{X/S})^2 / [K : Q] ) > Height_{Fal}(J(X_K)), where $K_{X/S}$ is the canonically…
In this paper we present new results about arrangements of lines and osculating curves associated to the Fermat curves in the projective plane. We first consider the sextactic points on the Fermat curves and show that they are distributed…
Motivated by asymptotic phenomena of moduli spaces of higher rank stable sheaves on algebraic surfaces, we study the Picard number of the moduli space of one-dimensional stable sheaves supported in a sufficiently positive divisor class on a…
In this paper, we consider rational cuspidal plane curves having at least three cusps. We give an upper bound of the self-intersection number of the proper transforms of such curves via the minimal embedded resolution of the cusps. For a…
We obtain a sharp bound on the number of self-intersections of a closed planar curve with trigonometric parameterization. Moreover, we show that a generic curve of this form is normal in the sense of Whitney.
We prove that on separated algebraic surfaces every coherent sheaf is a quotient of a locally free sheaf. This class contains many schemes that are neither normal, reduced, quasiprojective or embeddable into toric varieties. Our methods…
In this paper, we study the Picard group of the moduli space of semistable sheaves on a smooth quadric surface. We polarize the surface by an ample divisor close to the anticanonical class. We focus especially on moduli spaces of sheaves of…
We determine which of the modular curves $X_\Delta(N)$, that is, curves lying between $X_0(N)$ and $X_1(N)$, are bielliptic. Somewhat surprisingly, we find that one of these curves has exceptional automorphisms. Finally we find all…
A fast algorithm for counting intersections of two normal curves on a triangulated surface is proposed. It yields a convenient way for treating mapping class groups of punctured surfaces by presenting mapping classes by matrices, and the…
For two oriented simple closed curves on a compact orientable surface with a connected boundary we introduce a simple computation of a value in the first homology group of the surface, which detects in some cases that the geometric…
We study mapping class group orbits of homotopy and isotopy classes of curves with self-intersections. We exhibit the asymptotics of the number of such orbits of curves with a bounded number of self-intersections, as the complexity of the…
We prove new lower and upper bounds on the higher gonalities of finite graphs. These bounds are generalizations of known upper and lower bounds for first gonality to higher gonalities, including upper bounds on gonality involving…
The definition of the intersection number of a map with a closed manifold can be extended to the case of a closed stratified set such that the difference between dimensions of its two biggest strata is greater than $1$. The set Sigma of…