曲线及其割线簇的Clifford指数和行列式方程的推广
A generalization of the Clifford index and determinantal equations for curves and their secant varieties
摘要：
这是作者2008年在芝加哥大学的论文。我们推广的Clifford指数的概念，以任意的非常充分的线丛，并显示它如何确定时，曲线和它的各种割线品种有行列式方程。特别是我们建立了Eisenbud-Koh-Stillman猜想的概型理论版本。如果$L_1$和$L_2$是次数至少为$2g+1+k$的线丛，则对于$j \leq k$和对于$j \leq k+1$，如果$L_1 \neq L_2$，则确定地表示${\rm Sec}^j(C)$。给出了标准Clifford指标的几何刻画，并证明了$C$的正割簇在双正则嵌入下是行列式的.讨论了它与Koszul上同调和Green猜想的关系。
英文摘要：
This is the author's 2008 thesis from the University of Chicago. We generalize the notion of the Clifford index to an arbitrary very ample line bundle and show how it determines when a curve and its various secant varieties have determinantal equations. In particular we establish the scheme theoretic version of a conjecture of Eisenbud-Koh-Stillman. If $L_1$ and $L_2$ are line bundles of degree at least $2g+1+k$, then ${\rm Sec}^j(C)$ is determinantally presented for $j \leq k$ and for $j \leq k+1$ if $L_1 \neq L_2$. We also give a geometric characterization of the standard Clifford index in terms of which secant varieties of $C$ are determinantal in the bicanonical embedding. The relationship with Koszul cohomology and Green's conjecture is also discussed.
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