On projective normality and difining equations of a projective curve of genus three embedded by a complete linear system

Abstract

Introduction. Let $L : Cc*Ptl°a:>~1 be the projective embedding of a complete non-singular curve C of genus g by means of F(L), where L is a very ample invertible sheaf on C. We will study the homogeneous coordinate ring and the ideal of definitionI(L) of $L(C) in the case g=3. Our results are summarized in the following table. (If the genus of C is less than three, answers to the same kind of problems are easy.) In the table we will say that the homogeneous ideal I(L) is generated strictly by its elements of degrees vu ・・・,vm if I(L) is generated by its elements of degrees vlf・・■,vm and I(L) is not generated by its elements of degrees vu ・■■, vjt・・・, vm for any v, (l^j'^m), where 0, means that Vj is omitted.