On the G"ottsche Threshold
On the G"ottsche Threshold
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在Gottsche门槛上
DOI：
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发表时间：
2012
期刊：
影响因子：
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通讯作者：
with an appendix by Ilya Tyomkin
中科院分区：
文献类型：
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作者：
S. Kleiman;V. Shende;with an appendix by Ilya Tyomkin
对于光滑曲面S上的线丛L，如果L是d-非常充裕的，则上同格d曲线的Severi簇的次数由L和S的Chern类中的泛多项式给出。对于S有理数，我们大大放宽了后一个条件：只要有三个关键位点的余维数大于d就足够了。作为推论，我们证明了G“ottsche所设的条件在S是P^2或S是任意Hirzebruch曲面时成立，在S是任意经典del Pezzo曲面时也成立.
For a line bundle L on a smooth surface S, it is now known that the degree of the Severi variety of cogenus-d curves is given by a universal polynomial in the Chern classes of L and S if L is d-very ample. For S rational, we relax the latter condition substantially: it suffices that three key loci be of codimension more than d. As corollaries, we prove that the condition conjectured by G"ottsche suffices if S is P^2 or S is any Hirzebruch surface, and that a similar condition suffices if S is any classical del Pezzo surface.