Noncomplete embeddings of rational surfaces

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 $L\in PicX$ be a very ample line bundle. For a very ample subsystem $V\subset H^0(X,L)$ of codimension $t\ge 1$, if $X\hookrightarrow \P (V)$ satisfies Property $N_1^S$, then $Reg(X)\le t+2$\cite{KP}. Thus we investigate Property $N_1^S$ 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 $X\hookrightarrow \P (V)$ satisfies Property $N_1^S$. Indeed it is related to the geometry of a smooth rational curve of $X$. Also we apply our result to ${\P }^2$ and Hirzebruch surfaces.

8 pages