Noncomplete embeddings of rational surfaces
  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 $X \hookrightarrow \P (V)$ 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 $X \hookrightarrow \P (V)$ 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 $\P^2$ and Hirzebruch surfaces. 
Euisung Park