Enumeration of rational curves in a moving family of $\mathbb{P}^2$
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arXiv
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| Format: | Preprint |
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2018
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| _version_ | 1866913699734749184 |
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| author | Mukherjee, Ritwik Paul, Anantadulal Singh, Rahul Kumar |
| author_facet | Mukherjee, Ritwik Paul, Anantadulal Singh, Rahul Kumar |
| contents | We obtain a recursive formula for the number of rational degree $d$ curves in $\mathbb{P}^3$, whose image lies in a $\mathbb{P}^2$, passing through $r$ lines and $s$ points, where $r + 2s = 3d+2$. This can be viewed as a family version of the classical question of counting rational curves in $\mathbb{P}^2$. We verify that our numbers are consistent with those obtained by T. Laarakker, where he studies the parallel question of counting $δ$-nodal degree $d$ curves in $\mathbb{P}^3$ whose image lies inside a $\mathbb{P}^2$. Our numbers give evidence to support the conjecture, that the polynomials obtained by T. Laarakker are enumerative when $d \geq 1 + [\fracδ{2}]$, which is analogous to the {G}öttsche threshold for counting nodal curves in $\mathbb{P}^2$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1808_04237 |
| institution | arXiv |
| publishDate | 2018 |
| record_format | arxiv |
| spellingShingle | Enumeration of rational curves in a moving family of $\mathbb{P}^2$ Mukherjee, Ritwik Paul, Anantadulal Singh, Rahul Kumar Algebraic Geometry 14N35, 14J45 We obtain a recursive formula for the number of rational degree $d$ curves in $\mathbb{P}^3$, whose image lies in a $\mathbb{P}^2$, passing through $r$ lines and $s$ points, where $r + 2s = 3d+2$. This can be viewed as a family version of the classical question of counting rational curves in $\mathbb{P}^2$. We verify that our numbers are consistent with those obtained by T. Laarakker, where he studies the parallel question of counting $δ$-nodal degree $d$ curves in $\mathbb{P}^3$ whose image lies inside a $\mathbb{P}^2$. Our numbers give evidence to support the conjecture, that the polynomials obtained by T. Laarakker are enumerative when $d \geq 1 + [\fracδ{2}]$, which is analogous to the {G}öttsche threshold for counting nodal curves in $\mathbb{P}^2$. |
| title | Enumeration of rational curves in a moving family of $\mathbb{P}^2$ |
| topic | Algebraic Geometry 14N35, 14J45 |
| url | https://arxiv.org/abs/1808.04237 |