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Zbl 0603.14005
Levine, Marc N.; Srinivas, V.
Zero cycles on certain singular elliptic surfaces. (English)
[J] Compos. Math. 52, 179-196 (1984). ISSN 0010-437X; ISSN 1570-5846/e

Let Y be a smooth, projective surface over an algebraically closed field of characteristic $\ne 2$. Let $f: Y\to C$ be a morphism to a smooth curve whose general fibres are smooth elliptic curves. Let $P\sb 1,...,P\sb n$ be points of C such that $f\sp{-1}(P\sb i)$ is a reduced irreducible rational curve with a node $Q\sb i$. Let $X\to C$ be the singular projective surface obtained by blowing up Y at the $Q\sb i$ and then blowing down the strict transforms of the fibers $f\sp{-1}(P\sb i)$. Denote by $A\sb 0(X)$ the subgroup of the Grothendieck group $K\sb 0(X)$ generated by the sheaves with zero dimensional support contained in the smooth part of X. The main result is that the map $A\sb 0(X)\to A\sb 0(Y)$ is an isomorphism. Some examples of surfaces where the theorem can be applied are given.
[A.Parshin]

MSC 2000:: *14C25 Algebraic cycles
14C35 Appl. of methods of algebraic K-theory
14J17 Singularities of surfaces
14C05 Parametrization

Keywords: Chow groups; zero-zycles; Abel morphism; singular projective surface; Grothendieck group

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