论文标题
关于在给定程度的高度曲面中未包含的射曲线属的属
On the genus of projective curves not contained in hypersurfaces of given degree
论文作者
论文摘要
修复整数$ r \ geq 4 $和$ i \ geq 2 $（对于$ r = 4 $假设$ i \ geq 3 $）。假设等式$ \ binom {i+1} {i+1} {2} s+（i+1）= \ binom {r+i} {i} {i} $是一个整数，我们证明是一个较低和不可思议的复杂的项目cluve $ \ n $ \ p p^r p^r p^r p^r p^r p^r p^r p p， $ \ leq i $的高度曲面。事实证明，这种界限与Castelnuovo的界限相吻合，以$ \ MATHBB p^{s+1} $的曲线为曲线。我们证明，仅当存在一个整体表面$ s \ subset \ mathbb p^r $ s $时，限制是锋利的，而不是在$ \ leq i $的高度额外包含的。这样的表面（如果存在）必然是$ \ Mathbb p^{s+1} $的合理正常滚动表面的同构投影。这种表面$ S $的存在以$ i = 2 $和$ i = 3 $而闻名。因此，当$ i = 2 $或$ i = 3 $时，界限是锋利的，并且极端曲线是Castelnuovo的$ \ Mathbb p^r $的同构投影。
Fix integers $r\geq 4$ and $i\geq 2$ (for $r=4$ assume $i\geq 3$). Assuming that the rational number $s$ defined by the equation $\binom{i+1}{2}s+(i+1)=\binom{r+i}{i}$ is an integer, we prove an upper bound for the genus of a reduced and irreducible complex projective curve in $\mathbb P^r$, of degree $d\gg s$, not contained in hypersurfaces of degree $\leq i$. It turns out that this bound coincides with the Castelnuovo's bound for a curve of degree $d$ in $\mathbb P^{s+1}$. We prove that the bound is sharp if and only if there exists an integral surface $S\subset \mathbb P^r$ of degree $s$, not contained in hypersurfaces of degree $\leq i$. Such a surface, if existing, is necessarily the isomorphic projection of a rational normal scroll surface of degree $s$ in $\mathbb P^{s+1}$. The existence of such a surface $S$ is known for $i=2$ and $i=3$. It follows that, when $i=2$ or $i=3$, the bound is sharp, and the extremal curves are isomorphic projection in $\mathbb P^r$ of Castelnuovo's curves of degree $d$ in $\mathbb P^{s+1}$.
