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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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Cohomology algebra of plane curves, weak combinatorial type, and formality
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by J. I. Cogolludo Agustín and D. Matei PDF
Trans. Amer. Math. Soc. 364 (2012), 5765-5790 Request permission

Abstract:

We determine an explicit presentation by generators and relations of the cohomology algebra $H^*(\mathbb {P}^2\setminus \mathcal {C},\mathbb {C})$ of the complement to an algebraic curve $\mathcal {C}$ in the complex projective plane $\mathbb {P}^2$ via the study of log-resolution logarithmic forms on $\mathbb {P}^2$. As a first consequence, we derive that $H^*(\mathbb {P}^2\setminus \mathcal {C},\mathbb {C})$ depends only on the following finite pieces of data: the number of irreducible components of $\mathcal {C}$ together with their degrees and genera, the number of local branches of each component at each singular point, and the intersection numbers of every two distinct local branches at each singular point of $\mathcal {C}$. This finite set of data is referred to as the weak combinatorial type of $\mathcal {C}$. A further corollary is that the twisted cohomology jumping loci of $H^*(\mathbb {P}^2\setminus \mathcal {C},\mathbb {C})$ containing the trivial character also depend on the weak combinatorial type of $\mathcal {C}$. Finally, the explicit construction of the generators and relations allows us to prove that complements of plane projective curves are formal spaces in the sense of Sullivan.
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Additional Information
  • J. I. Cogolludo Agustín
  • Affiliation: Departamento de Matemáticas, IUMA, Universidad de Zaragoza, C/Pedro Cerbuna 12, CP50009 Zaragoza, Spain
  • Email: jicogo@unizar.es
  • D. Matei
  • Affiliation: Departamento de Matemáticas, Universidad de Zaragoza, C/Pedro Cerbuna 12, CP50009 Zaragoza, Spain – and – Institute of Mathematics “Simion Stoilow” of the Romanian Academy, P.O. Box 1-764, RO-014700 Bucharest, Romania
  • Email: daniel.matei@imar.ro
  • Received by editor(s): July 10, 2009
  • Received by editor(s) in revised form: October 14, 2010
  • Published electronically: June 22, 2012
  • Additional Notes: The first author was partially supported by the Spanish Ministry of Education MTM2010-21740-C02-02. The second author has been partially supported by SB2004-0181 and grant CNCSIS PNII-IDEI 1189/2008.
  • © Copyright 2012 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 364 (2012), 5765-5790
  • MSC (2010): Primary 14F25, 14F40, 14H50, 58A10, 58A12, 58A14, 14B05, 14E15, 32A27, 32S22, 55P62
  • DOI: https://doi.org/10.1090/S0002-9947-2012-05489-4
  • MathSciNet review: 2946931