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arXiv · math/0410309

Noncomplete embeddings of rational surfaces

Euisung Park
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Abstract

In this paper, we study the Castelnuovo-Mumford regularity of nonlinearly normal embedding of rational surfaces. Let $X$ be a rational surface and let $L \in {Pic}X$ be a very ample line bundle. For a very ample subsystem $V \subset H^0 (X,L)$ of codimension $t \geq 1$, if $X \hookrightarrow ¶(V)$ satisfies Property $N^S_1$, then ${Reg} (X) \leq t+2$\cite{KP}. Thus we investigate Property $N^S_1$ of noncomplete linear systems on X. And our main result is about a condition of the position of $V$ in $H^0 (X,L)$ such that $X \hookrightarrow ¶(V)$ satisfies Property $N^S_1$. Indeed it is related to the geometry of a smooth rational curve of $X$. Also we apply our result to $¶^2$ and Hirzebruch surfaces.

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Euisung Park. 2004-10-13. Noncomplete embeddings of rational surfaces. https://arxiv.org/abs/math/0410309

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