Some rational curves of maximal genus in $\mathbb{P}^3$
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For a reduced, irreducible and nondegenerate curve $C \subset \P^r$ of degree $d$, it was shown that the arithmetic genus $g$ of $C$ has an upper bound $\pi_0(d,r)$ by G. Castelnuovo. And he also classified the curves that attain the extremal value. These curves are arithmetically Cohen-Macaulay and contained in a surface of minimal degree. In this paper, we investigate the arithmetic genus of curves lie on a surface of minimal degree - the Veronese surface, smooth rational normal surface scrolls and singular rational normal surface scrolls. We also provide a construction of curves on singular rational normal surface scroll $S(0,2) \subset \P^3$ which attain the maximal arithmetic genus.
Abstract
For a reduced, irreducible and nondegenerate curve $C \subset \P^r$ of degree $d$, it was shown that the arithmetic genus $g$ of $C$ has an upper bound $\pi_0(d,r)$ by G. Castelnuovo. And he also classified the curves that attain the extremal value. These curves are arithmetically Cohen-Macaulay and contained in a surface of minimal degree. In this paper, we investigate the arithmetic genus of curves lie on a surface of minimal degree - the Veronese surface, smooth rational normal surface scrolls and singular rational normal surface scrolls. We also provide a construction of curves on singular rational normal surface scroll $S(0,2) \subset \P^3$ which attain the maximal arithmetic genus.
참고문헌 (12)
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