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Fractional intersection and bivariant theory. (English) Zbl 0774.14004
In Publ. Math., Inst. Hautes Étud. Sci. 9, 5-22 (1961; Zbl 0108.168), D. Mumford constructed an intersection number of two divisors on a normal surface, and the number can be a fractional number in general. The goal of this paper is to explain Mumford’s intersection using the bivariant intersection theory by Fulton and MacPherson [cf. W. Fulton, “Intersection theory” (1984; Zbl 0541.14005) and W. Fulton and R. MacPherson, Mem. Am. Math. Soc. 243 (1981; Zbl 0467.55005)].
For a variety $$X$$, let $$A_ *X$$ be its Chow group, namely cycles modulo rational equivalence. According to the bivariant theory, $$X$$ has the Chow cohomology $$A^*X$$ which is always a ring, and there is a natural map $$A^*X\to A_ *X$$, called the evaluation map. The evaluation map is bijective for smooth varieties, and it agrees with the usual intersection products. Similarly one can construct an algebraic equivalence version $$B^*X$$ together with the evaluation map to $$B_ *X$$. The main theorem of this paper says that when $$X$$ is a normal surface, the evaluation map $$B^*X\oplus\mathbb{Q}\to B_ *X\oplus\mathbb{Q}$$ is bijective, and it explains Mumford’s intersection, but the map $$A^*X\oplus\mathbb{Q}\to A_ *X\oplus\mathbb{Q}$$ is not always bijective.
To prove this, a method to compute $$A^*X$$, $$B^*X$$ and more general bivariant groups for singular varieties is studied.

##### MSC:
 14C15 (Equivariant) Chow groups and rings; motives 14C17 Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry 14C25 Algebraic cycles
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##### References:
 [1] Fulton, W. ”Intersection Theory”. · Zbl 0541.14005 [2] Fulton W., Mem. Amer. Math. Soc 243 (1981) [3] Mumford D., Publ. Math. LH.E.S 9 pp 5– (1961) [4] Vistoli A., Compositio, Math 70 pp 199– (1989) [5] DOI: 10.1007/BF01388892 · Zbl 0694.14001 · doi:10.1007/BF01388892 [6] DOI: 10.1112/S0025579300002102 · Zbl 0099.15902 · doi:10.1112/S0025579300002102
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