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The correction term for the Riemann–Roch formula of cyclic quotient singularities and associated invariants

Abstract

This paper deals with the invariant \(R_X\) called the RR-correction term, which appears in the Riemann–Roch and Numerical Adjunction Formulas for normal surface singularities. Typically, \(R_X=\delta ^\text {top}_X-\delta ^\text {an}_X\) decomposes as difference of topological and analytical local invariants of its singularities. The invariant \(\delta ^\text {top}_X\) is well understood and depends only on the dual graph of a good resolution. The purpose of this paper is to give a new interpretation for \(\delta ^\text {an}_X\), which in the case of cyclic quotient singularities can be explicitly computed via generic divisors. We also include two types of applications: one is related to the McKay decomposition of reflexive modules in terms of special reflexive modules in the context of the McKay correspondence. The other application answers two questions posed by Blache (Abh Math Semin Univ Hambg 65:307–340, 1995) on the asymptotic behavior of the invariant \(R_X\) of the pluricanonical divisor.

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Correspondence to José Ignacio Cogolludo-Agustín.

Additional information

Both authors are partially supported by the Spanish Government MTM2016-76868-C2-2-P and the Departamento de Industria e Innovación del Gobierno de Aragón and Fondo Social Europeo E22_17R Grupo Consolidado Álgebra y Geometría. The second author is also supported by FQM-333, from Junta de Andalucía.

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Cogolludo-Agustín, J.I., Martín-Morales, J. The correction term for the Riemann–Roch formula of cyclic quotient singularities and associated invariants. Rev Mat Complut 32, 419–450 (2019). https://doi.org/10.1007/s13163-018-0280-7

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  • DOI: https://doi.org/10.1007/s13163-018-0280-7

Keywords

  • Riemann-Roch
  • Adjunction formula
  • Cyclic quotient singularities
  • McKay correspondence
  • Reflexive modules
  • Curvettes

Mathematics Subject Classification

  • 32S05
  • 14H50
  • 32S25
  • 14F45