Journal of the Korean Mathematical Society (대한수학회지)

Korean Mathematical Society (대한수학회)
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CURVES WITH MAXIMAL RANK, BUT NOT ACM, WITH VERY HIGH GENERA IN PROJECTIVE SPACES

  • Received : 2018.10.08
  • Accepted : 2019.05.23
  • Published : 2019.09.01
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Abstract

A curve $X{\subset}\mathbb{P}^r$ has maximal rank if for each $t{\in}\mathbb{N}$ the restriction map $H^0(\mathcal{O}_{\mathbb{P}r}(t)){\rightarrow}H^0(\mathcal{O}_X(t))$ is either injective or surjective. We show that for all integers $d{\geq}r+1$ there are maximal rank, but not arithmetically Cohen-Macaulay, smooth curves $X{\subset}\mathbb{P}^r$ with degree d and genus roughly $d^2/2r$, contrary to the case r = 3, where it was proved that their genus growths at most like $d^{3/2}$ (A. Dolcetti). Nevertheless there is a sector of large genera g, roughly between $d^2/(2r+2)$ and $d^2/2r$, where we prove the existence of smooth curves (even aCM ones) with degree d and genus g, but the only integral and non-degenerate maximal rank curves with degree d and arithmetic genus g are the aCM ones. For some (d, g, r) with high g we prove the existence of reducible non-degenerate maximal rank and non aCM curves $X{\subset}\mathbb{P}^r$ with degree d and arithmetic genus g, while (d, g, r) is not realized by non-degenerate maximal rank and non aCM integral curves.

Keywords

Acknowledgement

Supported by : MIUR, GNSAGA of of INdAM (Italy)

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