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Curves with maximal rank, but not aCM, with very high genera in projective spaces
Edoardo Ballico 대한수학회 2019 대한수학회지 Vol.56 No.5
A curve $X\subset \mathbb {P}^r$ has maximal rank if for each $t\in \mathbb {N}$ the restriction map $H^0(\mathcal {O} _{\mathbb {P}^r}(t)) \to H^0(\mathcal {O} _X(t))$ is either injective or surjective. We show that for all integers $d\ge r+1$ 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.
Dependent subsets of embedded projective varieties
Edoardo Ballico 대한수학회 2020 대한수학회보 Vol.57 No.4
Let $X\subset \mathbb {P}^r$ be an integral and non-degenerate variety. Set $n:= \dim (X)$. Let $\rho (X)''$ be the maximal integer such that every zero-dimensional scheme $Z\subset X$ smoothable in $X$ is linearly independent. We prove that $X$ is linearly normal if $\rho (X)''\ge \lceil (r+2)/2\rceil$ and that $\rho (X)'' < 2\lceil (r+1)/(n+1)\rceil$, unless either $n=r$ or $X$ is a rational normal curve.
Footnote to a manuscript by Gwena and Teixidor i Bigas
Edoardo Ballico,Claudio Fontanari 대한수학회 2009 대한수학회보 Vol.46 No.1
Recent work by Gwena and Teixidor i Bigas provides a characteristic-free proof of a part of a previous theorem by one of us, under a stronger numerical assumption. By using an intermediate result from the mentioned manuscript, here we present a simpler, characteristic-free proof of the whole original statement.