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摘要翻译：
为了完成几乎极小度的类，即度精确超过余维数2的非退化不可约射影类的分类理论和结构理论，一个自然的方法是研究极小度的类的简单射影。设$\tilde X\子集{\mathbb P}^{r+1}_k$为各种最小度且余维度至少为2,并考虑$x_p=\pi_p(\tilde X)\子集{\mathbb P}^r_k$中的$P\在{\mathbb P}^{r+1}_k\反斜杠\tilde X$。通过引用{B-Sche}证明了$x_p$的上同调和局部性质由$\tildex$关于$p$的割线轨迹$\sigma_p(\tildex)$控制。沿着这一思路，本文给出了$tilde x$的割线分层的几何描述，即${mathbb P}^{r+1}_k$通过割线轨迹类型的分解。我们证明了割线轨迹$\sigma_p(\tildex)$有六种可能，并精确地描述了$\tildex$割线分层的每一层，每一层都是拟射影变体。作为一个应用，我们通过提供一个完整的对$(\tilde X,p)$,其中$\tilde X\子集{\mathbb p}^{r+1}_k$是一个极小度的变种，$p$是$\mathbb p}^{r+1}_k\setminus\tilde X$中的一个闭点，$x_p\子集{\mathbb p}^r_k$是一个Del Pezzo变种，得到了所有非正规Del Pezzo变种的分类。
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英文标题：
《On varieties of almost minimal degree I: Secant loci of rational normal
scrolls》
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作者：
M. Brodmann and E. Park
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最新提交年份：
2009
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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：
To complete the classification theory and the structure theory of varieties of almost minimal degree, that is of non-degenerate irreducible projective varieties whose degree exceeds the codimension by precisely 2, a natural approach is to investigate simple projections of varieties of minimal degree. Let $\tilde X \subset {\mathbb P}^{r+1}_K$ be a variety of minimal degree and of codimension at least 2, and consider $X_p = \pi_p (\tilde X) \subset {\mathbb P}^r_K$ where $p \in {\mathbb P}^{r+1}_K \backslash \tilde X$. By \cite{B-Sche}, it turns out that the cohomological and local properties of $X_p$ are governed by the secant locus $\Sigma_p (\tilde X)$ of $\tilde X$ with respect to $p$. Along these lines, the present paper is devoted to give a geometric description of the secant stratification of $\tilde X$, that is of the decomposition of ${\mathbb P}^{r+1}_K$ via the types of secant loci. We show that there are exactly six possibilities for the secant locus $\Sigma_p (\tilde X)$, and we precisely describe each stratum of the secant stratification of $\tilde X$, each of which turns out to be a quasi-projective variety. As an application, we obtain the classification of all non-normal Del Pezzo varieties by providing a complete list of pairs $(\tilde X, p)$ where $\tilde X \subset {\mathbb P}^{r+1}_K$ is a variety of minimal degree, $p$ is a closed point in $\mathbb P^{r+1}_K \setminus \tilde X$ and $X_p \subset {\mathbb P}^r _K$ is a Del Pezzo variety.
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PDF链接：
https://arxiv.org/pdf/0808.0090
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