Let X be a projective surface over C with an isolated singularity and let S to X be its desingularization. Assuming that the exceptional divisor C is irreducible and nonsingular, the author constructs a regulator map :H1(X,K2) to H/H2(S −C,Z)(1), where H is the dual of the kernel of the residue map H1(S,1 S(2C))!H1(C,O). This map generalizes the regulator map for nonsingular surfaces and is expected to provide information about the singularity of X. This point is illustrated by the example of the ordinary quadratic cone, where the author computes that (KernelH1(X,K2)!H1(S,K2)) is isomorphic to C.
A regulator map for surfaces with an isolated singularity.
COLLINO, Alberto
1991-01-01
Abstract
Let X be a projective surface over C with an isolated singularity and let S to X be its desingularization. Assuming that the exceptional divisor C is irreducible and nonsingular, the author constructs a regulator map :H1(X,K2) to H/H2(S −C,Z)(1), where H is the dual of the kernel of the residue map H1(S,1 S(2C))!H1(C,O). This map generalizes the regulator map for nonsingular surfaces and is expected to provide information about the singularity of X. This point is illustrated by the example of the ordinary quadratic cone, where the author computes that (KernelH1(X,K2)!H1(S,K2)) is isomorphic to C.
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