Let X be a smooth projective variety of dimension n in P^r. We study the fibers of a general linear projection pi: X --> P^{n+c}, with c > 0. When n is small it is classical that the degree of any fiber is bounded by n/c+1, but this fails for n >> 0. We describe a new invariant of the fiber that agrees with the degree in many cases and is always bounded by n/c+1. This implies, for example, that if we write a fiber as the disjoint union of schemes Y' and Y'' such that Y' is the union of the locally complete intersection components of Y, then deg Y'+deg Y''_red <= n/c+1 and this formula can be strengthened a little further. Our method also gives a sharp bound on the subvariety of P^r swept out by the l-secant lines of X for any positive integer l, and we discuss a corresponding bound for highly secant linear spaces of higher dimension. These results extend Ziv Ran's "Dimension+2 Secant Lemma".
本文研究了Segre簇的割线簇的维数。这个问题既可以在张量代数的背景下求解，也可以在代数几何的背景下求解。一个归纳过程是建立在点和投影的连续专业化的想法。这将高维情况下割线簇的维数的计算减少到低维情况下部分割线簇的一系列计算。作为应用的技术：我们给一个完整的分类缺陷$t$正割品种Segre品种t < 7。推广了Catalisano-Geramita-Gimigliano关于P^n的张量幂幂的非亏性定理.我们确定了一套p的不平衡Segre品种有缺陷的p-割线品种。此外，我们还证明了Segre簇P^1 x P^1 x P^n x P^n和P^2 x P^3 x P^3是亏的，并完整地描述了它们的割线簇的维数。在最后一节中，我们提出了一系列关于缺陷型Segre品种的建议。
On the irreducibility of secant cones, and an application to linear normality
设$Y \subset ¶^r$是正规非退化m维子簇，$σ(Y)$表示子簇$Z \subset Y_{smooth}$的最大维数，使得$Z$包含$Y$上某个因子的类属点，且所有$y \in Z$的切平面$T_y Y$包含在一个固定超平面内.本文研究了双位点$D \subset $ Y $ of its generic projection to $ <$^{r-1} $, proving that if the secant variety of $ Y $ is the whole space and $ σ（Y）<2 m-r + 1 $, then $ D $ is irreducible. Applying Zak's Tangency theorem we deduce the irreducibility of $ D $ when $ m > 2（r-1）/3$.后者意味着Zak线性正态性定理的一个版本。
