The Maximum Genus Problem for Locally Cohen-Macaulay Space Curves

Abstract

Let \({P_{\rm MAX}(d, s)}\) denote the maximum arithmetic genus of a locally Cohen-Macaulay curve of degree d in \({\mathbb{P}^3}\) that is not contained in a surface of degree < s. A bound P(d, s) for \({P_{\rm MAX}(d, s)}\) has been proven by the first author in characteristic zero and then generalized in any characteristic by the third author. In this paper, we construct a large family \({\mathcal{C}}\) of primitive multiple lines and we conjecture that the generic element of \({\mathcal{C}}\) has good cohomological properties. From the conjecture it would follow that \({P(d, s) = P_{\rm MAX}(d, s)}\) for d = s and for every \({d \geq 2s - 1}\). With the aid of Macaulay2 we checked this holds for \({s \leq 120}\) by verifying our conjecture in the corresponding range.

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Correspondence to Paolo Lella.

Additional information

This research is supported by MIUR funds PRIN 2015 project Geometria delle varietà algebriche (coordinator A. Verra) and by MIUR funds FFABR-BEORCHIA-2018. All authors are members of GNSAGA.

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Beorchia, V., Lella, P. & Schlesinger, E. The Maximum Genus Problem for Locally Cohen-Macaulay Space Curves. Milan J. Math. 86, 137–155 (2018). https://doi.org/10.1007/s00032-018-0284-2

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Mathematics Subject Classification (2010)

  • Primary: 14C05
  • Secondary: 14H50
  • 14Q05
  • 13P10

Keywords

  • Hilbert scheme
  • locally Cohen-Macaulay curve
  • initial ideal
  • weight vector
  • Gröbner basis
  • smooth curve