摘要翻译：
本文研究了一些射影变体的简单线性射影在射影中心穿过周围空间时的行为。更准确地说，设$X\子集\p^r$是一个满足Green-Lazarsfeld性质$n_p$的射影簇，对于某个$p\geq2$,$q\在\p^r$外的闭点，并且$x_q:=\pi_q(X)\子集\p^{r-1}$来自$q$的$X$的投影图像。首先证明了$X$关于$q$的割线轨迹$\sigma_q(X)$,即$X$上所有点跨越通过$q$的割线的集合，在$\p^r$的子空间中要么是空的，要么是二次的。这意味着有限态射$\pi_q:X\to x_q$是双形的。我们的主要结果是，$x_q$的上同调和局部性质精确地由$\sigma_q(X)$确定。为了完成这一结果，下一步应该对所有可能的割线轨迹进行分类，并通过割线轨迹的分类来分解周围空间。我们得到了Veronese嵌入和Segre嵌入的这样一个分解。作为主要结果的应用，我们研究了低次簇的上同调性质。
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英文标题：
《On secant loci and simple linear projections of some projective
  varieties》
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作者：
Euisung Park
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最新提交年份：
2008
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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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一级分类：Mathematics        数学
二级分类：Commutative Algebra        交换代数
分类描述：Commutative rings, modules, ideals, homological algebra, computational aspects, invariant theory, connections to algebraic geometry and combinatorics
交换环，模，理想，同调代数，计算方面，不变理论，与代数几何和组合学的联系
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英文摘要：
  In this paper, we study how simple linear projections of some projective varieties behave when the projection center runs through the ambient space. More precisely, let $X \subset \P^r$ be a projective variety satisfying Green-Lazarsfeld's property $N_p$ for some $p \geq 2$, $q \in \P^r$ a closed point outside of $X$, and $X_q := \pi_q (X) \subset \P^{r-1}$ the projected image of $X$ from $q$. First, it is shown that the secant locus $\Sigma_q (X)$ of $X$ with respect to $q$, i.e. the set of all points on $X$ spanning secant lines passing through $q$, is either empty or a quadric in a subspace of $\P^r$. This implies that the finite morphism $\pi_q : X \to X_q$ is birational. Our main result is that cohomological and local properties of $X_q$ are precisely determined by $\Sigma_q (X)$. To complete this result, the next step should be to classify all possible secant loci and to decompose the ambient space via the classification of secant loci. We obtain such a decomposition for Veronese embeddings and Segre embeddings. Also as an application of the main result, we study cohomological properties of low degree varieties.
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PDF链接：
https://arxiv.org/pdf/0808.2005