论文标题
在$ \ mathbb {p}^4 $ of度量$ d = g+1 $的希尔伯特方案中
On the Hilbert scheme of smooth curves in $\mathbb{P}^4$ of degree $d = g+1$ and genus $g$ with negative Brill-Noether number
论文作者
论文摘要
我们用$ \ Mathcal {h} _ {d，g，r} $光滑曲线的希尔伯特方案，该方案是组件的结合，其一般点对应于平稳的不可减至且非降级曲线的$ d $ $ d $和属$ g $ g $ in $ \ pp^r $。在本文中，我们表明，对于Brill-Noether系列以外的低属$ g $，Hilbert方案$ \ Mathcal {h} _ {g+1，g，4} $是非空的，只要$ g \ ge 9 $和不可降低，唯一的组件通常会呈线性正常曲线，除非有线性正常的曲线，否则$ g = 9 $ = 9 $或$ g = 9 $或$ g = 12 $ = 12 $。这补充了Severi原始断言的有效性，即$ \ Mathcal {h} _ {d，g，g，r} $在Brill-nover范围之外的$ d = g+1 $和$ r = 4 $之外的不可约性。
We denote by $\mathcal{H}_{d,g,r}$ the Hilbert scheme of smooth curves, which is the union of components whose general point corresponds to a smooth irreducible and non-degenerate curve of degree $d$ and genus $g$ in $\PP^r$. In this article, we show that for low genus $g$ outside the Brill-Noether range, the Hilbert scheme $\mathcal{H}_{g+1,g,4}$ is non-empty whenever $g\ge 9$ and irreducible whose only component generically consists of linearly normal curves unless $g=9$ or $g=12$. This complements the validity of the original assertion of Severi regarding the irreducibility of $\mathcal{H}_{d,g,r}$ outside the Brill-Nother range for $d=g+1$ and $r=4$.
