Inequalities between the Chern numbers of a singular fiber in a family of algebraic curves
Inequalities between the Chern numbers of a singular fiber in a family of algebraic curves
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DOI：
10.1090/s0002-9947-2012-05625-x
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发表时间：
2010-03
影响因子：
1.3
通讯作者：
Jun Lu;Shengli Tan
中科院分区：
文献类型：
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作者：
Jun Lu;Shengli Tan
曲线族中一根奇异纤维的陈氏数是该纤维对整个空间整体陈氏数的局部贡献。本文的第一个目的是找出奇异纤维的陈氏数之间的最佳不等式。我们的第二个目的是试图给出一种新的方法来分类亏格g。我们知道当g很大时，亏格g的奇异纤维太多而不能完全分类(见[5]，[7]，[8]，[16])。为了得到不变量之间的局部-全局关系，一种可能的方法是根据奇异纤维对全局不变量的贡献对其进行分类。为了解释这一方法，我们将对具有大或小陈数的奇异纤维进行分类，并给出一些应用。有关曲线族的局部-全局性质的研究背景，请参阅调查[2]。C上的亏格g的曲线族是一个纤维f：X→C，它的一般纤维F是亏格g的光滑曲线，其中X是复杂光滑射影曲面。如果所有的奇异纤维都是约化的节点曲线，则称这个族为半稳定的。如果X=F×C，且f正好是C的第二个投影，则称f为平凡族。如果f的所有光滑纤维彼此同构，等价地，f在有限的基变化C→C下变平凡，则称f为等平凡的。我们总是假设f是相对极小的，即在任何奇异纤维中都不存在(−1)曲线。当g=1时，Kodaira[6]从奇异纤维中找到了全局不变量。第一陈氏数c1(X)始终为零，第二陈氏数c2(X)通过Noether公式等于12χ(Ox)，以及
Chern numbers of a singular fiber in a family of curves are the local contributions of the fiber to the global Chern numbers of the total space. Our first purpose of this paper is to find the best inequalities between the Chern numbers of a singular fiber. Our second purpose is to try to give a new approach to the classification of singular fibers of genus g. We know that when g is big, there are too many singular fibers of genus g to classify completely (see [5], [7], [8], [16]). In order to get the local-global relations between the invariants, one possible way is to classify singular fibers according to their contributions to the global invariants. To explain this approach, we will classify singular fibers with big or small Chern numbers and give some applications. See the survey [2] for the background of the study on the local-global properties for families of curves. A family of curves of genus g over C is a fibration f : X → C whose general fibers F are smooth curves of genus g, where X is a complex smooth projective surface. The family is called semistable if all of the singular fibers are reduced nodal curves. If X = F ×C and f is just the second projection to C, then we call f a trivial family. If all of the smooth fibers of f are isomorphic to each other, equivalently, f becomes trivial under a finite base change C → C, then f is called isotrivial. We always assume that f is relatively minimal, i.e., there is no (−1)-curve in any singular fiber. When g = 1, Kodaira [6] found the global invariants from the singular fibers. The first Chern number c1(X) is always zero, the second Chern number c2(X) is equal to 12χ(OX) by Noether’s formula, and