论文标题
Coxeter理论用于$ \ Mathbb {p}^r $的爆炸的曲线理论
Coxeter theory for curves on blowups of $\mathbb{P}^r$
论文作者
论文摘要
我们研究了对$ y_s^r $平稳的不可约理性曲线的研究，这是$ \ mathbb {p}^r $ at $ s $一般积分的一般爆炸，其正常捆绑包为直接的线条总和$ i $ $ $ i $，用于$ i \ in \ in \ in \ in \ in \ in \ in \ { - 1,1,0,1,1,1 \} $：$ curves $ curves。我们系统地利用了在$ y_s^r $中应用于曲线的Chow空间的Coxeter组的理论，该理论为我们提供了有用的双线形式，有助于公开$（i）$ - 曲线的属性。我们对标准Cremona转换的Weyl组（所有这些都是$（i）$ - 曲线）下的线条轨道（通过$ 1-i $点）特别感兴趣的：我们称这些$（i）$ - Weyl Lines。我们证明了与理解$（i）$ - 曲线是$（i）$ - Weyl Line有关的各种定理，这是通过双线性形式表示的数值标准。我们获得了$ r = 3 $的更强结果，在此我们证明了符合尖锐标准的Noether-type不平等现象。
We investigate the study of smooth irreducible rational curves in $Y_s^r$, a general blowup of $\mathbb{P}^r$ at $s$ general points, whose normal bundle splits as a direct sum of line bundles all of degree $i$, for $i \in \{-1,0,1\}$: we call these $(i)$-curves. We systematically exploit the theory of Coxeter groups applied to the Chow space of curves in $Y_s^r$, which provides us with a useful bilinear form that helps to expose properties of $(i)$-curves. We are particularly interested in the orbits of lines (through $1-i$ points) under the Weyl group of standard Cremona transformations (all of which are $(i)$-curves): we call these $(i)$-Weyl lines. We prove various theorems related to understanding when an $(i)$-curve is an $(i)$-Weyl line, via numerical criteria expressed in terms of the bilinear form. We obtain stronger results for $r=3$, where we prove a Noether-type inequality that gives a sharp criterion.
