摘要翻译：
在自然假设下，通过对Hilb^2(S)中有理曲线相关族的研究，给出了曲面上p_g(S)>0的超椭圆正规奇异曲线族维数的一个上界。利用这一结果，在一般的K3曲面上证明了具有超椭圆规格化的几何亏格3的节点曲线的存在性，从而在其Hilbert平方上得到了特定的二维有理曲线族。我们描述了两个无穷级数的一般基元极化K3的例子，使得它们的Hilbert平方包含一个IP^2或K3上一个IP^1-丛的三倍双形。讨论了一般K3的Hilbert平方的Mori锥的一些结果。
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英文标题：
《On families of rational curves in the Hilbert square of a surface (with
  an Appendix by Edoardo Sernesi)》
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作者：
Flaminio Flamini, Andreas Leopold Knutsen, Gianluca Pacienza, Edoardo
  Sernesi
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最新提交年份：
2007
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分类信息：

一级分类：Mathematics        数学
二级分类：Algebraic Geometry        代数几何
分类描述：Algebraic varieties, stacks, sheaves, schemes, moduli spaces, complex geometry, quantum cohomology
代数簇，叠，束，格式，模空间，复几何，量子上同调
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英文摘要：
  Under natural hypotheses we give an upper bound on the dimension of families of singular curves with hyperelliptic normalizations on a surface S with p_g(S) >0 via the study of the associated families of rational curves in Hilb^2(S).   We use this result to prove the existence of nodal curves of geometric genus 3 with hyperelliptic normalizations, on a general K3 surface, thus obtaining specific 2-dimensional families of rational curves in its Hilbert square. We describe two infinite series of examples of general, primitively polarized K3's such that their Hilbert squares contain a IP^2 or a threefold birational to a IP^1-bundle over a K3.   We discuss some consequences on the Mori cone of the Hilbert square of a general K3.
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PDF链接：
https://arxiv.org/pdf/0704.1367