论文标题
$ \ mathbb p^4 $和$ \ mathbb p^5 $中的曲线属不包含在Quadrics中
The genus of curves in $\mathbb P^4$ and $\mathbb P^5$ not contained in quadrics
论文作者
论文摘要
投影曲线理论中的一个经典问题是将其所有可能属的分类从嵌入的空间的程度和维度上分类。固定的整数$ r，d，s $，Castelnuovo-Halphen的理论指出，在不包含在$ <s $ $ <s $的表面上的条件下，在$ \ mathbb p^r $的非分级，降低和不可减少曲线的属上均具有$ D $ $ d $的不可约D $。该理论可以通过多种方式推广。例如，固定的整数$ r，d，k $，可能会要求一个曲线的最大属$ d $ in $ \ mathbb p^r $，但不包含在$ <k $的高度表面中。在本文中，我们研究了Quadrics中未包含的曲线$ c $ c $ c $ in $ \ mathbb p^r $（即$ h^0（\ mathbb p^r，\ mathcal i_c（2））= 0 $）。当$ r = 4 $和$ r = 5 $以及$ d \ gg0 $时，我们为该属展示了尖锐的上限。对于$ r \ geq 7 $的某些值，我们能够确定除恒定术语以外的锐角，并且该参数也适用于立方体中不包含的曲线。
A classical problem in the theory of projective curves is the classification of all their possible genera in terms of the degree and the dimension of the space where they are embedded. Fixed integers $r,d,s$, Castelnuovo-Halphen's theory states a sharp upper bound for the genus of a non-degenerate, reduced and irreducible curve of degree $d$ in $\mathbb P^r$, under the condition of being not contained in a surface of degree $<s$. This theory can be generalized in several ways. For instance, fixed integers $r,d,k$, one may ask for the maximal genus of a curve of degree $d$ in $\mathbb P^r$, not contained in a hypersurface of a degree $<k$. In the present paper we examine the genus of curves $C$ of degree $d$ in $\mathbb P^r$ not contained in quadrics (i.e. $h^0(\mathbb P^r, \mathcal I_C(2))=0$). When $r=4$ and $r=5$, and $d\gg0$, we exhibit a sharp upper bound for the genus. For certain values of $r\geq 7$, we are able to determine a sharp bound except for a constant term, and the argument applies also to curves not contained in cubics.
