摘要翻译：
本文比较了$\pp^n$上的自由扭丛$\ff与具有相同秩和分裂类型的自由向量丛$\oplus_{i=1}^n\opn(b_i)$。我们证明了第一个总是“少”的全局部分，而它有一个更高的第二个Chern类。在这两种情况下，根据最大自由子束$\FF$找到了差值的界。因此，我们得到了“Horrocks分裂判据”的一个直接、简单和更一般的证明，它对无扭转槽也成立，并且仅依赖于$\ff$的一些数值不变量，就得到了$\ff$的Chern类$C_i(\ff(t))$的下界。特别地，对于分裂型没有间隙的$pp^n$上的秩$n$无扭束($b_i\geq b_{i+1}\geq b_i-1$对于每$i=1,...,n-1$)证明了以下判别公式:\[\\delta(\ff):=2nc_2-(n-1)c_1^2\geq-{1/12}n^2(n^2-1)\]最后，在秩$n$自反束的情况下，我们得到了上同调模的维数的高Chern类$c_3(\ff(t))，...，c_n(\ff(t))$的绝对值的多项式上界\FF(t)$和Castelnuovo-Mumford正则性为$\FF$；这些多项式界仅依赖于$C_1(\ff)$，$C_2(\ff)$，$\ff$和$T$的分裂类型。
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英文标题：
《Splitting type, global sections and Chern classes for torsion free
  sheaves on P^N》
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作者：
Cristina Bertone, Margherita Roggero
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最新提交年份：
2010
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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：
  In this paper we compare a torsion free sheaf $\FF$ on $\PP^N$ and the free vector bundle $\oplus_{i=1}^n\OPN(b_i)$ having same rank and splitting type. We show that the first one has always "less" global sections, while it has a higher second Chern class. In both cases bounds for the difference are found in terms of the maximal free subsheaves of $\FF$. As a consequence we obtain a direct, easy and more general proof of the "Horrocks' splitting criterion", also holding for torsion free sheaves, and lower bounds for the Chern classes $c_i(\FF(t))$ of twists of $\FF$, only depending on some numerical invariants of $\FF$. Especially, we prove for rank $n$ torsion free sheaves on $\PP^N$, whose splitting type has no gap (i.e. $b_i\geq b_{i+1}\geq b_i-1$ for every $i=1, ...,n-1$), the following formula for the discriminant: \[ \Delta(\FF):=2nc_2-(n-1)c_1^2\geq -{1/12}n^2(n^2-1)\] Finally in the case of rank $n$ reflexive sheaves we obtain polynomial upper bounds for the absolute value of the higher Chern classes $c_3(\FF(t)), ..., c_n(\FF(t))$, for the dimension of the cohomology modules $H^i\FF(t)$ and for the Castelnuovo-Mumford regularity of $\FF$; these polynomial bounds only depend only on $c_1(\FF)$, $c_2(\FF)$, the splitting type of $\FF$ and $t$.
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PDF链接：
https://arxiv.org/pdf/0804.2985