Degenerations of complete collineations and geometric Tevelev degrees of $\mathbb{P}^r$
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2023
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| _version_ | 1866909303739252736 |
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| author | Lian, Carl |
| author_facet | Lian, Carl |
| contents | We consider the problem of enumerating maps $f$ of degree $d$ from a fixed general curve $C$ of genus $g$ to $\mathbb{P}^r$ satisfying incidence conditions of the form $f(p_i)\in X_i$, where $p_i\in C$ are general points and $X_i\subset\mathbb{P}^r$ are general linear spaces. We give a complete answer in the case where the $X_i$ are points, where the counts, the ``Tevelev degrees'' of $\mathbb{P}^r$, were previously known only when $r=1$, when $d$ is large compared to $r,g$, or virtually in Gromov-Witten theory. We also give a complete answer in the case $r=2$ with arbitrary incidence conditions. Our main approach studies the behavior of complete collineations under various degenerations. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2308_00046 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Degenerations of complete collineations and geometric Tevelev degrees of $\mathbb{P}^r$ Lian, Carl Algebraic Geometry Combinatorics We consider the problem of enumerating maps $f$ of degree $d$ from a fixed general curve $C$ of genus $g$ to $\mathbb{P}^r$ satisfying incidence conditions of the form $f(p_i)\in X_i$, where $p_i\in C$ are general points and $X_i\subset\mathbb{P}^r$ are general linear spaces. We give a complete answer in the case where the $X_i$ are points, where the counts, the ``Tevelev degrees'' of $\mathbb{P}^r$, were previously known only when $r=1$, when $d$ is large compared to $r,g$, or virtually in Gromov-Witten theory. We also give a complete answer in the case $r=2$ with arbitrary incidence conditions. Our main approach studies the behavior of complete collineations under various degenerations. |
| title | Degenerations of complete collineations and geometric Tevelev degrees of $\mathbb{P}^r$ |
| topic | Algebraic Geometry Combinatorics |
| url | https://arxiv.org/abs/2308.00046 |