## LSU Historical Dissertations and Theses

#### Title

Weierstrass Points on Gorenstein Curves.

1984

Dissertation

#### Degree Name

Doctor of Philosophy (PhD)

Mathematics

#### Abstract

This thesis generalizes the classical definition of a Weierstrass point to integral projective Gorenstein curves. For X an integral projective Gorenstein curve of arithmetic genus g at least two, pick P(epsilon)X, and call the sheaf of dualizing differentials (omega). For a proper closed subscheme Z of X with support P and ideal sheaf I, define the degree of Z to be dim(,(//C)) O(,P)/I(,P). Call Z 1-special if dim(,(//C))Hom(I,O(,X)) (GREATERTHEQ) 1. In a manner analogous to the classical construction, one defines the wronskian of X to be some element (alpha) of (H('0) (X,(omega)('(CRTIMES) 1/2(g('2)+g)))). One defines the Weierstrass weight W(P) to be ord(,P) (alpha), and one calls P a Weierstrass point of X if W(P) > 0. Then. Theorem 1. The following statements are equivalent for P(epsilon)X. (1) W(P) > 0. (2) There is a nonzero (sigma)(epsilon)H('0)(X,(omega)) satisfying ord(,P) (sigma) (GREATERTHEQ) g. (3) There is a 1-special subscheme with support P and length equal to g. (4) There is a 1-special subscheme with support P and length at most g. Consider the following statements about P(epsilon)X. (A) There is a morphism (phi): X (--->) ('1) of degree at most g satisfying (phi)('-1)((phi)(P)) = P . (B) There is a principal 1-special subscheme with support P and degree at most g. (C) P is a Weierstrass point. If P is nonsingular (A) (B) (C). If P is singular, then one only has (A) (IMPLIES) (B) (IMPLIES) (C). But, there is the following. Theorem 2. Let P(epsilon)X(,sing) satisfy (delta)(,P) = 1. (1) If P is a cusp then P satisfies (A). (2) If P is a node then P satisfies (C). P does not satisfy (B) if and only if (i) X is of genus three; (ii) Let (theta): Y (--->) X be the normalization at P with (theta)('-1)(P) = Q,R . Then Q and R are Weierstrass points of Y.

97

#### DOI

10.31390/gradschool_disstheses.3998

COinS