Coxeter theory for curves on blowups of $\mathbb{P}^r$
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arXiv
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| Format: | Preprint |
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2022
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| _version_ | 1866910049472872448 |
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| author | Dumitrescu, Olivia Miranda, Rick |
| author_facet | Dumitrescu, Olivia Miranda, Rick |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2205_13605 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Coxeter theory for curves on blowups of $\mathbb{P}^r$ Dumitrescu, Olivia Miranda, Rick Algebraic Geometry 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. |
| title | Coxeter theory for curves on blowups of $\mathbb{P}^r$ |
| topic | Algebraic Geometry |
| url | https://arxiv.org/abs/2205.13605 |