Divisor

About: Divisor is a research topic. Over the lifetime, 2462 publications have been published within this topic receiving 21394 citations. The topic is also known as: factor & submultiple.

Linear independence in linear systems on elliptic curves

Abstract: Let $E$ be an elliptic curve, with identity $O$, and let $C$ be a cyclic subgroup of odd order $N$, over an algebraically closed field $k$ with $\operatorname{char} k mid N$. For $P \in C$, let $s_P$ be a rational function with divisor $N \cdot P - N \cdot O$. We ask whether the $N$ functions $s_P$ are linearly independent. For generic $(E,C)$, we prove that the answer is yes. We bound the number of exceptional $(E,C)$ when $N$ is a prime by using the geometry of the universal generalized elliptic curve over $X_1(N)$. The problem can be recast in terms of sections of an arbitrary degree $N$ line bundle on $E$.

Residues and obstructions for log-symplectic manifolds

Abstract: We consider compact Kahlerian manifolds $X$ of even dimension 4 or more, endowed with a log-symplectic structure $\Phi$, a generically nondegenerate closed 2-form with simple poles on a divisor $D$ with local normal crossings. A simple linear inequality involving the iterated Poincare residues of $\Phi$ at components of the double locus of $D$ ensures that the pair $(X, \Phi)$ has unobstructed deformations and that $D$ deforms locally trivially.

On common index divisors and monogenity of certain number fields defined by $x^5+ax^2+b$.

Abstract: Let $K=\Q(\alpha)$ be a number field generated by a complex root $\alpha$ of a monic irreducible trinomial $F(x) = x^5+ax^2+b \in \Z[x]$. In this paper, for every prime integer $p$, we give necessary and sufficient conditions on $a$ and $b$ so that $p$ is a common index divisor of $K$. In particular, when these conditions hold, then $K$ is not monogenic.

The Incidence Variety Compactification of strata of d-differentials in genus 0

Abstract: Given $d\in \mathbb{Z}_{\geq 2}$, for every $\underline{k}=(k_1,\dots,k_n) \in \mathbb{Z}^{n}$ such that $k_i\geq 1-d$ and $k_1+\dots+k_n=-2d$, denote by $\Omega^d\mathcal{M}_{0,n}(\underline{k})$ and $\mathbb{P}\Omega^d\mathcal{M}_{0,n}(\underline{k})$ the corresponding stratum of $d$-differentials in genus $0$ and its projectivization respectively. We show that the incidence variety compactification of $\mathbb{P}\Omega^d\mathcal{M}_{0,n}(\underline{k})$ is isomorphic to the blow-up of $\overline{\mathcal{M}}_{0,n}$ along a specific sheaf of ideals. Along the way we obtain an explicit divisor representing the tautological line bundle on the incidence variety. In the case where none of the $k_i$ is divisible by $d$, the self-intersection number of this divisor computes the volume of $\mathbb{P}\Omega^d\mathcal{M}_{0,n}(\underline{k})$. We prove a recursive formula which allows one to compute the volume of $\mathbb{P}\Omega^d\mathcal{M}_{0,n}(\underline{k})$ from the volumes of other strata of lower dimensions.

A lower bound for $\chi (\mathcal O_S)$

Abstract: Let $(S,\mathcal L)$ be a smooth, irreducible, projective, complex surface, polarized by a very ample line bundle $\mathcal L$ of degree $d > 25$. In this paper we prove that $\chi (\mathcal O_S)\geq -\frac{1}{8}d(d-6)$. The bound is sharp, and $\chi (\mathcal O_S)=-\frac{1}{8}d(d-6)$ if and only if $d$ is even, the linear system $|H^0(S,\mathcal L)|$ embeds $S$ in a smooth rational normal scroll $T\subset \mathbb P^5$ of dimension $3$, and here, as a divisor, $S$ is linearly equivalent to $\frac{d}{2}Q$, where $Q$ is a quadric on $T$. Moreover, this is equivalent to the fact that the general hyperplane section $H\in |H^0(S,\mathcal L)|$ of $S$ is the projection of a curve $C$ contained in the Veronese surface $V\subseteq \mathbb P^5$, from a point $x\in V\backslash C$.