On the Generators of the First Homology with Compact Supports of the Weierstrass Family in Characteristic Zero
نویسنده
چکیده
Let WQ = Proj(Q[g2' g3' X, Y, ZJI(homogeneous ideal generated by y 2Z + 4X3 g2XZ2 g3Z3». This is said to be the Weierstrass Family over the field Q. Then the first homology with compact supports of the Weierstrass Family is computed explicitly, i.e., it is generated by {C-kdX 1\ dYh .. , and {XC-kdX 1\ dYh .. 1 over the ring Q[g2' g3]' where C is a polynomial y 2 4X3 + g2 X + g3' When one tensors the homology of the Weierstrass Family with 1l-'Q[g2' g3], being localized at the discriminant Il = gi 27g?, over Q[g2' g3], the first homology is generated by C-1dX 1\ dY and XC-1dX 1\ dY. One also obtains the first homologies with compact supports of singular fibres over flO = (g2 = g3 = 0) and flO = (g2 = 3, g3 = I) as corollaries. Introduction. We wish to compute the Q[g2' g3]-adic homology with compact supports of the Weierstrass Family WQ, where ·( Q[g2' g3' X, Y, z] ) W = Pr~ . Q homogeneous ideal generated by y 2Z + 4X3 g2XZ2 g3Z3 We regard the graded ring Q[g2' g3' X, Y, Z] as the graded Q[g2' g3]-algebra such that X, Yand Z each has degree + I and all the elements of Q[g2' g3] have degree zero. Let U be the open subset of WQ, "the finite points": U = WQ n A2(Spec(Q[g2' g3]))' This is the closed subscheme of A2(Spec(Q[g2' g3])) given by y 2 = 4X3 g2X g3' Then we have the long exact sequence of the homology with compact supports, ... -> H%_2({points at oo}, Q[g2' g3]) -> H%(WQ , Q[g2' g3]) -> H~'(U, Q[g2' g3]) -> .••. Since H%({points at oo}, Q[g2' g3]) vanishes except at h = 0, we have h =1= 2, h = 2. Therefore the knowledge of H%(U, Q[g2' g3])' h;;;. 0, determines the homology groups of all the fibres in the family over the various points ~ E Spec(Q[g2,g3])' i.e., E;,q = Tor~[g2,g3)(H~(U,Q[g2' g3],K(~») Received by the editors June 8, 1982. The contents of this paper have been presented to the American Mathematical Society Meeting held at Monterey, California, November 19, 1982. 1980 Mathematics Subject Classification. Primary 14GIO, 14F30; Secondary IOBIO, 14K07,
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