Daniel Naie
A formula for the jumping numbers of a curve unibranch at a singular point is established. The jumping numbers are expressed in terms of the Enriques diagram of the log resolution of the singularity, or equivalently in terms of the canonical set of generators of the semigroup of the curve at the singular point.
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We investigate the jumping conics of stable vector bundles $E$ of rank 2 on a smooth quadric surface $Q$ with the first Chern class $c_1=\Oo_Q(-1,-1)$ with respect to the ample line bundle $\Oo_Q(1,1)$. We show that the set of jumping conics of $E$ is a hypersurface of degree $c_2(E)-1$ in $\PP_3^*$. Using these hypersurfaces, we describe moduli spaces of stable vector bundles of rank 2 on $Q$ in the cases of lower $c_2(E)$.
We define Schwarzenberger bundles on any smooth projective variety X. We introduce the notions of jumping pairs of a Steiner bundle E on X and determine a bound for the dimension of its jumping locus. We completely classify Steiner bundles whose set of jumping pairs have maximal dimension, proving that they are all Schwarzenberger bundles.
We give an example that the volume of an $\mathbb{R}$-divisor on a family of complex smooth surfaces jumps at infinite many prime divisors in the base. Our example follows the construction in \cite{1}.
In this article, we study the jumping numbers of an ideal in the local ring at rational singularity on a complex algebraic surface. By understanding the contributions of reduced divisors on a fixed resolution, we are able to present an algorithm for finding of the jumping numbers of the ideal. This shows, in particular, how to compute the jumping numbers of a plane curve from the numerical data of its minimal resolution. In addition, the jumping numbers of the maximal ideal at the singular point in a Du Val or toric surface singularity are computed, and applications to the smooth case are expl…
Given a singular curve on a smooth surface, we determine which exceptional divisors on the minimal resolution of that curve contribute toward its jumping numbers.
Multiplicities of Jumping Numbers
本文研究任意维光滑簇中乘子理想跳跃数的重数，证明重数函数为拟多项式，从而庞加莱级数有理，并探讨最高次数分量与里斯赋值及单项式理想情形。
We investigate the jumping conics of stable vector bundles $\Ee$ of rank 2 on a smooth quadric surface $Q$ with the Chern classes $c_1=\Oo_Q(-1,-1)$ and $c_2=4$ with respect to the ample line bundle $\Oo_Q(1,1)$. We describe the set of jumping conics of $\Ee$, a cubic symmetroid in $\PP_3$, in terms of the cohomological properties of $\Ee$. As a consequence, we prove that the set of jumping conics, $S(\Ee)$, uniquely determines $\Ee$. Moreover we prove that the moduli space of such vector bundles is rational.
Codimension of jumping loci
本文研究光滑射影簇上向量丛限制到曲线族时Harder-Narasimhan过滤的变化，确定了斜率跳跃轨迹余维数线性依赖跳跃量的几何条件，并应用于研究射影平面秩2丛及曲线模空间奇点轨迹。
Given a smooth complex algebraic variety $X$ and a nonzero regular function $f$ on $X$, we give an effective estimate for the difference between the jumping numbers of $f$ and the $F$-jumping numbers of a reduction $f_p$ of $f$ to characteristic $p\gg 0$, in terms of the roots of the Bernstein-Sato polynomial $b_f$ of $f$. As an application, we show that if $b_f$ has no roots of the form $-{\rm lct}(f)-n$, with $n$ a positive integer, then the $F$-pure threshold of $f_p$ is equal to the log canonical threshold of $f$ for $p\gg 0$ with $(p-1){\rm lct}(f)\in {\mathbf Z}$.
For a simple complete ideal $\wp$ of a local ring at a closed point on a smooth complex algebraic surface, we introduce an algebraic object, named Poincaré series $P_{\wp}$, that gathers in an unified way the jumping numbers and the dimensions of the vector space quotients given by consecutive multiplier ideals attached to $\wp$. This paper is devoted to prove that $P_{\wp}$ is a rational function giving an explicit expression for it.
We study in this paper some local invariants attached via multiplier ideals to an effective divisor or ideal sheaf on a smooth complex variety. First considered (at least implicitly) by Libgober and by Loeser and Vaquie, these jumping coefficients consist of an increasing sequence of rational numbers beginning with the log canonical threshold of the divisor or ideal in question. They encode interesting geometric and algebraic information, and we show that they arise naturally in several different contexts. Given a polynomial f having only isolated singularities, results of Varchenko, Loeser an…
Understanding when an abstract complex curve of given genus comes equipped with a map of fixed degree to a projective space of fixed dimension is a foundational question; and Brill--Noether theory addresses this question via linear series, which algebraically codify maps to projective targets. Classical Brill--Noether theory, which focuses on smooth curves, has been intensively explored; but much less is known for singular curves, particularly for those with non-nodal singularities. In a one-parameter family of smooth curves specializing to a singular curve $C_0$, one expects certain aspects o…
Let $X$ denote an integral, projective Gorenstein curve over an algebraically closed field $k$. In the case when $k$ is of characteristic zero, C. Widland and the second author have defined Weierstrass points of a line bundle on $X$. In the first section, this definition is extended to linear systems in arbitrary characteristic. This definition may be viewed as a generalization of the definitions of Laksov and Stöhr-Voloch to the Gorenstein case. In the second section, an example is given to illustrate the definition. This example is a plane curve of arithmetic genus 3 in characteristic 2 such…
Multiplier ideals of plane curve singularities via Newton polygons
本文提出一种有效方法，利用 Newton 多边形及环面嵌入解消确定光滑曲面曲线奇点的乘子理想与跳跃数，并推广了 Howald 定理。
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