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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.
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참고문헌 (17)
참고문헌 신청
Alberto Alzati / 2002 / On the k-normality of projected algebraic varieties / Bulletin of the Brazilian Mathematical Society, New Series 33(1):27~48
E Ballico / 1989 / On projections of ruled and veronese surfaces / Journal of Algebra 121(2):477~487
Edoardo Ballico / 2018 / Maximal rank of space curves in the range A / European Journal of Mathematics 4(3):778~801
Markus Brodmann / 2007 / Arithmetic properties of projective varieties of almost minimal degree / Journal of Algebraic Geometry 16(2):347~400
I. A. Chel’tsov / 1997 / Del Pezzo surfaces with nonrational singularities / Mathematical Notes 62(3):377~389
E Ballico / 1989 / On projections of ruled and veronese surfaces / Journal of Algebra 121(2):477~487
Edoardo Ballico / 2018 / Maximal rank of space curves in the range A / European Journal of Mathematics 4(3):778~801
Markus Brodmann / 2007 / Arithmetic properties of projective varieties of almost minimal degree / Journal of Algebraic Geometry 16(2):347~400
I. A. Chel’tsov / 1997 / Del Pezzo surfaces with nonrational singularities / Mathematical Notes 62(3):377~389
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