Weierstrass points on Gorenstein curves (Q1112897)

The authors investigate which equivalent definitions for Weierstrass points on smooth curves remain equivalent on Gorenstein curves. It is shown that a Wronskian definition using dualizing differentials is equivalent to a definition involving 1-special subschemes with support at a point. The appearance of 1-special subschemes, instead of just special divisors, is to be expected since the limit of a divisor may be a subscheme, but not a divisor, as smooth curves degenerate to a singular curve. Let X denote an integral, projective Gorenstein curve of arithmetic genus \(g>1.\) An example is given to show that there may not be a locally principal 1-special subscheme (i.e. special Cartier divisor) of degree at most g at a (singular) Weierstrass point P on X. Another example is given to show that there may not exist a morphism \(\phi: X\to {\mathbb{P}}^ 1\) of degree at most g such that \(\phi^{-1}(\phi (P))=\{P\}\).