Pietro De Poi
In this article we obtain a complete description of the congruences of lines in $\p^4$ of order one provided that the fundamental surface $F$ is non-reduced (and possibly reducible) at one of its generic points, and their classification under the hypothesis that $(F)_{\red}$ is smooth.
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On generically non-reduced components of Hilbert schemes of smooth curves
本文利用Hodge理论技术，构造了光滑曲线Hilbert概型的泛非约化分支的新例子，突破了超曲面次数限制，并获得了某些Hodge轨迹的泛非约化分支。
We explore some equisingularity criteria in one parameter families of generically reduced curves. We prove the equivalence between Whitney regularity and Zariski's discriminant criterion. We prove that topological triviality implies smoothness of the normalized surface. Examples are given to show that Witney regularity and equisaturation are not stable under the blow-up of the singular locus nor under the Nash modification.
The Jacobi Bound Conjecture for Generically Reduced Differential Schemes
本文证明了微分方案一般约简成分的强雅可比界猜想，通过核心方法解决了相关数学问题。
The volume of a line bundle is defined in terms of a limsup. It is a fundamental question whether this limsup is a limit. It has been shown that this is always the case on generically reduced schemes. We show that volumes are limits in two classes of schemes that are not necessarily generically reduced: codimension one subschemes of projective varieties such that their components of maximal dimension contain normal points and projective schemes whose nilradical squared equals zero.
In this paper we give some criteria for a family of generically reduced plane curve singularities to be equinormalizable. The first criterion is based on the $δ$-invariant of a (non-reduced) curve singularity which is introduced by Brücker-Greuel (\cite{BG}). The second criterion is based on the I-equisingularity of a $k$-parametric family ($k\geq 1$) of generically reduced plane curve singularities, which is introduced by Nobile (\cite{No}) for one-parametric families. The equivalence of some kinds of equisingularities of a family of generically reduced plane curve singularities is also studi…
In this article, we study the Hilbert scheme of generically non-reduced curves in $\mathbb{P}^3$. We prove the existence of generically non-reduced curves in $\mathbb{P}^3$ for which there exist infinitesimal deformations of the curve that do not induce deformations of the associated reduced scheme. We show that such infinitesimal deformations contribute to the non-reducedness of the corresponding Hilbert scheme. We introduce the notion of extension of curves and prove that such infinitesimal deformations (such that the associated reduced scheme does not deform) are inherited by the extended c…
The present paper concerns the invariants of generically nef vector bundles on ruled surfaces. By Mehta - Ramanathan Restriction Theorem and by Miyaoka characterization of semistable vector bundles on a curve, the generic nefness can be considered as a weak form of semistability. We establish a Bogomolov type inequality for generically nef vector bundles with nef general fiber restriction on ruled surfaces with no negative section. This gives an affermative answer in this case to a problem posed by Th. Peternell. Concerning ruled surfaces with a negative section, we prove a a similar result fo…
In the present paper we describe new component of the Gieseker-Maruyama moduli space $\mathcal{M}(14)$ of coherent semistable rank-2 sheaves with Chern classes $c_1=0, \ c_2=14, \ c_3=0$ on $\mathbb{P}^{3}$ which is generically non-reduced. The construction of this component is based on the technique of elementary transformations of sheaves and famous Mumford's example of a non-reduced component of the Hilbert scheme of smooth space curves of degree 14 and genus 24.
In this note we consider the moduli space of stable bundles of rank two on a very general quintic surface. We study the potentially obstructed points of the moduli space via the spectral covering of a twisted endomorphism. This analysis leads to generically non-reduced components of the moduli space, and components which are generically smooth of more than the expected dimension. We obtain a sharp bound asked for by O'Grady saying when the moduli space is good.
We study maximal families W of the Hilbert scheme, H(d,g)_{sc}, of smooth connected space curves whose general curve C lies on a smooth surface S of degree s. We give conditions on C under which W is a generically smooth component of H(d,g)_{sc} and we determine dim W. If s=4 and W is an irreducible component of H(d,g)_{sc}, then the Picard number of S is at most 2 and we explicitly describe, also for s > 4, non-reduced and generically smooth components in the case Pic(S) is generated by the classes of a line and a smooth plane curve of degree s-1. For curves on smooth cubic surfaces the first…
We give a sufficient condition for a first order infinitesimal deformation of a curve on a 3-fold to be obstructed. As application we construct generically non-reduced components of the Hilbert schemes of uniruled 3-folds and the Hom scheme from a general curve of genus five to a general cubic 3-fold.
We study families of ropes of any codimension that are supported on lines. In particular, this includes all non-reduced curves of degree two. We construct suitable smooth parameter spaces and conclude that all ropes of fixed degree and genus lie in the same component of the corresponding Hilbert scheme. We show that this component is generically smooth if the genus is small enough unless the characteristic of the ground field is two and the curves under consideration have degree two. In this case the component is non-reduced.
We continue the study of maximal families W of the Hilbert scheme, H(d,g)_{sc}, of smooth connected space curves whose general curve C lies on a smooth degree-s surface S containing a line. For s > 3, we extend the two ranges where W is a unique irreducible (resp. generically smooth) component of H(d,g)_{sc}. In another range, close to the boarder of the nef cone, we describe for s=4 and 5 components W that are non-reduced, leaving open the non-reducedness of only 3 (resp. 2) families for s > 5 (resp. s=5), thus making progress to recent results of Kleppe and Ottem in [28]. For s=3 we slightly…
Characterizing Jacobians via the KP equation and via flexes and degenerate trisecants to the Kummer variety: an algebro-geometric approach
本文因定理4证明存在缺陷而撤回，该定理断言阿贝尔簇上丰富线丛完全线性系统的基轨迹既约；其余关于用KP方程和Kummer簇三割线刻画雅可比簇的代数几何证明正确，但依赖该定理的较弱版本。
The Delta Invariant and Simultaneous Normalization for Families of Isolated Non-Normal Singularities
本文研究了孤立非正常奇点的delta不变量及其在族中的上半连续性，提出在纤维上可规范化的条件，并推广了经典结果。
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