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• W2035332801 abstract "It is shown that as a family of projective smooth curves degenerates to an irreducible Gorenstein curve the Weierstrass weight at a P on the limit curve is the sum of the Weierstrass weights at points on the smooth curves converging to P. C. Widland [8] extended the classical notion of Weierstrass to integral, projective Gorenstein curves. We considered Weierstrass points of invertible sheaves on such curves in [5] and showed that a singular is always a Weierstrass of high weight of any invertible sheaf with at least two linearly independent global sections. We remarked that this may be interpreted to mean that as a family of smooth curves degenerates to an irreducible Gorenstein curve, then many of the Weierstrass points tend towards the singularities. Our goal in this note is to make this statement precise. We work over C. By a point of a scheme, we mean a closed point. If X is a scheme of finite type over C, then we denote the associated complex space by Xan. We wish to thank Augusto Nobile and Peter Stiller for helpful conversations. Let S be an integral, noetherian scheme over C and let ir: X -S be a family of Gorenstein curves of arithmetic genus g > 2. By this we will mean that 7r is a flat, proper morphism whose geometric fibers are integral Gorenstein curves of arithmetic genus g. By the theory of duality of coherent sheaves [4, V.9.7 and VII.4 (p. 388)], there is a canonical invertible sheaf w = wx/s on X whose restriction to a fiber X, is the sheaf of dualizing differentials on that curve. Let L be an invertible sheaf on X. Then L is flat over S. Assume that dim h0(X,, Ls) = r > 0 for all s E S. Then 7r2 is locally free of rank r. We define below an effective Cartier divisor 1 (L) on X. We will also use 1 (L) to denote the associated locally principal closed subscheme of X. We note that the subscheme W(L) may not be reduced. The points of W(L) are the L-Weierstrass points of X/S. The wOn-Weierstrass points are called Weierstrass points of order n and the w-Weierstrass points are simply called Weierstrass points. Suppose (s, P) E X and let U be an open neighborhood of (s, P) such that (1) There are sectionsl, ,. . ., Or of L whose restrictions to X,, form a basis for HO(X5,, , L) for all s' e ir(U). (2) LC(U) is a free Ox (U)-module generated by /. (3) wx/s (U) is a free Ox (U)-module generated by r. Define F%,j E F(U, OX) by q$3 = F1,> for j=,...,r, dFi-1,3 = Ft,,3 for i = 2, . ,r and j = 1,. . . ., r, Received by the editors May 15, 1986 and, in revised form, June 27, 1986. 1980 Mathematics Subject Classification (1985 Revision). Primary 14H15." @default.
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• W2035332801 date "1987-01-01" @default.
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• W2035332801 title "Weierstrass weight and degenerations" @default.
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• W2035332801 doi "https://doi.org/10.1090/s0002-9939-1987-0897062-8" @default.
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