Projective Normality Of Algebraic Curves And Its Application To Surfaces

Abstract

Let $L$ be a very ample line bundle on a smooth curve $C$ of genus $g$ with $\frac{3g+3}{2}<°L\le 2g-5$. Then $L$ is normally generated if $°L>\max \{2g+2-4h^1(C,L),2g-\frac{g-1}{6}-2h^1(C,L)\}$. Let $C$ be a triple covering of genus $p$ curve $C^{\prime }$ with $C\stackrel{\varphi }{\to }{C}^{\prime }$ and $D$ a divisor on $C^{\prime }$ with $4p<°D<\frac{g-1}{6}-2p$. Then $K_C(-{\varphi }^{\ast }D)$ becomes a very ample line bundle which is normally generated. As an application, we characterize some smooth projective surfaces.

7 pages, 1figure