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.

Birational models of ${\mathcal M}_{2,2}$ arising as moduli of curves with nonspecial divisors

Abstract: We study birational projective models of ${\mathcal M}_{2,2}$ obtained from the moduli space of curves with nonspecial divisors. We describe geometrically which singular curves appear in these models and show that one of them is obtained by blowing down the Weierstrass divisor in the moduli stack of ${\mathcal Z}$-stable curves $\overline{\mathcal M}_{2,2}({\mathcal Z})$ defined by Smyth. As a corollary, we prove projectivity of the coarse moduli space $\overline{M}_{2,2}({\mathcal Z})$.

Ext and Tor on two-dimensional cyclic quotient singularities

Abstract: Given two torus invariant Weil divisors $D$ and $D'$ on a two-dimensional cyclic quotient singularity $X$, the groups $\mathop{Ext} olimits^i_{X}(\mathcal{O}(D),\mathcal{O}(D'))$, $i>0$, are naturally $\mathbb{Z}^2$-graded. We interpret these groups via certain combinatorial objects using methods from toric geometry. In particular, it is enough to give a combinatorial description of the $\mathop{Ext} olimits^1$-groups in the polyhedra of global sections of the Weil divisors involved. Higher $\mathop{Ext} olimits^i$-groups are then reduced to the case of $\mathop{Ext} olimits^1$ via a quiver. We use this description to show that $\mathop{Ext} olimits^1_{X}(\mathcal{O}(D),\mathcal{O}(K-D')) = \mathop{Ext} olimits^1_{X}(\mathcal{O}(D'),\mathcal{O}(K-D))$, where $K$ denotes the canonical divisor on $X$. Furthermore, we show that $\mathop{Ext} olimits^{i+2}_{X}(\mathcal{O}(D),\mathcal{O}(D'))$ is the Matlis dual of $\mathop{Tor} olimits_{i}^{X}(\mathcal{O}(D),\mathcal{O}(D'))$.

On a Restricted Divisor Problem

Abstract: Let 0 < α < 1/2 and let d α (n) be the number of positive divisors k of n such that n α ≤ k ≤ n 1-α , which we call a restricted divisor function. In the case α = 1/N (N ∈ N) we derive an asymptotic representation of Σn≤x d α (n) . Furthermore we study the mean square of P α (x) = Σ l≤x α φ (x/l), which seems to be a natural object in the case of a restricted divisor problem.

Free divisors and their deformations

Abstract: A reduced divisor D = V (f) Cn is free if the sheaf Der(-logD) := f 2 DerCn (f) 2 (f)OCng of logarithmic vector fields is a locally free OCn-module. It is linear if, furthermore, Der(-logD) is globally generated by a basis consisting of vector fields all of whose coefficients, with respect to the standard basis @=@x1;...; @=@xn of the space DerCn of vector fields on Cn, are linear functions. In principle, linear free divisors, like other kinds of singularities, might be expected to appear in non-trivial parameterised families. As part of this thesis, however, we prove that for reductive linear free divisors, there are no formally non-trivial families, where a linear free divisor is reductive if its associated Lie algebra is reductive, thus reductive linear free divisors are formally rigid. To prove this and to understand better the class of free divisors, we introduce a rigorous deformation theory for germs of free and linear free divisors. A (linearly) admissible deformation is a deformation in which we deform a germ of a (linear) free divisor (D; 0) c (Cn; 0) in such a way that each fiber of the deformation is still a (linear) free divisor and that the singular locus of (D; 0) is deformed atly. Moreover, we explain how to use the de Rham logarithmic complex to compute the space of first order infinitesimal admissible deformations and the Lie algebra cohomology complex to compute the space of first order infinitesimal linearly admissible deformations.

Pseudo-effectivity of the relative canonical divisor and uniruledness in positive characteristic

Abstract: We show that if $f: X \to T$ is a surjective morphism between smooth projective varieties over an algebraically closed field $k$ of characteristic $p>0$ with geometrically integral and non-uniruled generic fiber, then $K_{X/T}$ is pseudo-effective. The hardest part of the proof is to show that there is a finite smooth non-uniruled cover of the base, for which we show the following: if $T$ is a smooth projective variety over $k$, and $\mathcal{A}$ is an ample enough line bundle, then a cyclic cover of degree $p mid d$ given by a general element of $\left|\mathcal{A}^d\right|$ is not uniruled. For this we show the following cohomological uniruledness condition, which might be of independent interest: a smooth projective variety $T$ of dimenion $n$ is not uniruled whenever the dimension of the semi-stable part of $H^n(T, \mathcal{O}_T)$ is greater than that of $H^{n-1}(T, \mathcal{O}_T)$. Additionally we also show singular versions of all the above statements.