Limits of special Weierstrass points

Abstract

Let C be the union of two general connected, smooth, nonrational curves X and Y intersecting transversally at a point P. Assume that P is a general point of X or of Y. Our main result, in a simplified way, says: Let Q be a point of X. Then Q is the limit of special Weierstrass points on a family of smooth curves degenerating to C if and only if Q is not P and either of the following conditions hold: Q is a special ramification point of the linear system |K_X+(g_Y+1)P|, or Q is a ramification point of the linear system |K_X+(g_Y+1+j)P| for j=-1 or j=1 and P is a Weierstrass point of Y. Above, g_Y stands for the genus of Y and K_X for a canonical divisor of X. As an application, we recover in a unified and conceptually simpler way computations made by Diaz and Cukierman of certain divisor classes in the moduli space of stable curves. In our method there is no need to worry about multiplicities, an usual nuisance of the method of test curves.