Rational cuspidal curves in a moving family of ℙ2

In this paper we obtain a formula for the number of rational degree d curves in ℙ3 having a cusp, whose image lies in a ℙ2 and that passes through r lines and s points (where r + 2s = 3d + 1). This problem can be viewed as a family version of the classical question of counting rational cuspidal curv...

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Autores principales:	Mukherjee Ritwik, Singh Rahul Kumar
Formato:	article
Lenguaje:	EN
Publicado:	De Gruyter 2021
Materias:	euler class degenerate contributions enumerative geometry 14n35 14j45 Mathematics QA1-939
Acceso en línea:	https://doaj.org/article/49a62da795d84af69ee68f656593c985
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spelling	oai:doaj.org-article:49a62da795d84af69ee68f656593c9852021-12-05T14:10:45ZRational cuspidal curves in a moving family of ℙ22300-744310.1515/coma-2020-0110https://doaj.org/article/49a62da795d84af69ee68f656593c9852021-02-01T00:00:00Zhttps://doi.org/10.1515/coma-2020-0110https://doaj.org/toc/2300-7443In this paper we obtain a formula for the number of rational degree d curves in ℙ3 having a cusp, whose image lies in a ℙ2 and that passes through r lines and s points (where r + 2s = 3d + 1). This problem can be viewed as a family version of the classical question of counting rational cuspidal curves in ℙ2, which has been studied earlier by Z. Ran ([13]), R. Pandharipande ([12]) and A. Zinger ([16]). We obtain this number by computing the Euler class of a relevant bundle and then finding out the corresponding degenerate contribution to the Euler class. The method we use is closely based on the method followed by A. Zinger ([16]) and I. Biswas, S. D’Mello, R. Mukherjee and V. Pingali ([1]). We also verify that our answer for the characteristic numbers of rational cuspidal planar cubics and quartics is consistent with the answer obtained by N. Das and the first author ([2]), where they compute the characteristic number of δ-nodal planar curves in ℙ3 with one cusp (for δ ≤ 2).Mukherjee RitwikSingh Rahul KumarDe Gruyterarticleeuler classdegenerate contributionsenumerative geometry14n3514j45MathematicsQA1-939ENComplex Manifolds, Vol 8, Iss 1, Pp 125-137 (2021)
institution	DOAJ
collection	DOAJ
language	EN
topic	euler class degenerate contributions enumerative geometry 14n35 14j45 Mathematics QA1-939
spellingShingle	euler class degenerate contributions enumerative geometry 14n35 14j45 Mathematics QA1-939 Mukherjee Ritwik Singh Rahul Kumar Rational cuspidal curves in a moving family of ℙ2
description	In this paper we obtain a formula for the number of rational degree d curves in ℙ3 having a cusp, whose image lies in a ℙ2 and that passes through r lines and s points (where r + 2s = 3d + 1). This problem can be viewed as a family version of the classical question of counting rational cuspidal curves in ℙ2, which has been studied earlier by Z. Ran ([13]), R. Pandharipande ([12]) and A. Zinger ([16]). We obtain this number by computing the Euler class of a relevant bundle and then finding out the corresponding degenerate contribution to the Euler class. The method we use is closely based on the method followed by A. Zinger ([16]) and I. Biswas, S. D’Mello, R. Mukherjee and V. Pingali ([1]). We also verify that our answer for the characteristic numbers of rational cuspidal planar cubics and quartics is consistent with the answer obtained by N. Das and the first author ([2]), where they compute the characteristic number of δ-nodal planar curves in ℙ3 with one cusp (for δ ≤ 2).
format	article
author	Mukherjee Ritwik Singh Rahul Kumar
author_facet	Mukherjee Ritwik Singh Rahul Kumar
author_sort	Mukherjee Ritwik
title	Rational cuspidal curves in a moving family of ℙ2
title_short	Rational cuspidal curves in a moving family of ℙ2
title_full	Rational cuspidal curves in a moving family of ℙ2
title_fullStr	Rational cuspidal curves in a moving family of ℙ2
title_full_unstemmed	Rational cuspidal curves in a moving family of ℙ2
title_sort	rational cuspidal curves in a moving family of ℙ2
publisher	De Gruyter
publishDate	2021
url	https://doaj.org/article/49a62da795d84af69ee68f656593c985
work_keys_str_mv	AT mukherjeeritwik rationalcuspidalcurvesinamovingfamilyofp2 AT singhrahulkumar rationalcuspidalcurvesinamovingfamilyofp2
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Rational cuspidal curves in a moving family of ℙ2

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