An algebraic approach to the intersection theory.

*(English)*Zbl 0599.14003
Curves Semin. at Queen’s, Vol. 2, Kingston/Can. 1981-82, Queen’s Pap. Pure Appl. Math. 61, Exp. A, 32 p. (1982).

Summary: [For the entire collection see Zbl 0584.00014.]

The paper provides an algebraic approach to intersection theory by studying a formula for the product of degrees of two intersecting varieties \(V\) and \(W\) in terms of algebraic data, in particular when \(V\) and \(W\) are subvarieties of the projective space. The basis for the formula is a method of expressing the intersection multiplicity of two properly intersecting varieties as the length of a canonically associated primary ideal. This basic idea for the calculation of multiplicities occurs explicitly in papers published by S. Kleiman and by the authors in 1972. It is an open problem in the paper to describe the connection between this algebraic approach and Fulton’s intersection theory. In the meantime this is given in the following papers: L. van Gastel: ”A geometric approach to Vogel’s intersection theory” (Preprint, Univ. Utrecht, The Netherlands, September 1984) and W. Fulton: ”Algebraic refined intersections” (Preprint, Brown Univ., Providence, R.I., November 1984).

The paper provides an algebraic approach to intersection theory by studying a formula for the product of degrees of two intersecting varieties \(V\) and \(W\) in terms of algebraic data, in particular when \(V\) and \(W\) are subvarieties of the projective space. The basis for the formula is a method of expressing the intersection multiplicity of two properly intersecting varieties as the length of a canonically associated primary ideal. This basic idea for the calculation of multiplicities occurs explicitly in papers published by S. Kleiman and by the authors in 1972. It is an open problem in the paper to describe the connection between this algebraic approach and Fulton’s intersection theory. In the meantime this is given in the following papers: L. van Gastel: ”A geometric approach to Vogel’s intersection theory” (Preprint, Univ. Utrecht, The Netherlands, September 1984) and W. Fulton: ”Algebraic refined intersections” (Preprint, Brown Univ., Providence, R.I., November 1984).