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Explicit computations in \(Hilb^ 3\,{\mathbb{P}}^ 2\). (English) Zbl 0656.14005

Algebraic geometry, Proc. Conf., Sundance/Utah 1986, Lect. Notes Math. 1311, 76-100 (1988).
[For the entire collection see Zbl 0635.00006.]
The authors consider the Hilbert scheme \(Hilb^ 3(P^ 2)\) which parametrizes the triplets in the projective plane, and its ring of rational equivalence CH. Their aim is to describe the multiplicative structure of this ring and to give some enumerative applications. The details and other applications are announced in two other papers.
The authors introduce the incidence variety \(Hilb^ 3(P^ 3)\) of pairs (t,d) where d is a doublet and t a triplet containing d. They prove that this variety is smooth. No such result is known for a general \(Hilb^ d(P^ 2)\), although \(Hilb^ d(P^ 2)\) is known to be smooth (Hartshorne, Fogarty). Next the authors recall the computation of the additive structure of CH by Ellingsrud and Strømme, and then they give explicit generators of the ring by cycles having a geometric meaning and prove that the \(CH^ i\) are free groups generated by monominals on these cycles.
The paper ends with three enumerative applications: number of curves on a net \({\mathcal N}\) osculating a given curve or having a singular point on it, number of singular points appearing in two given nets.
Reviewer: J.M.Granger

MSC:

14C05 Parametrization (Chow and Hilbert schemes)
14C15 (Equivariant) Chow groups and rings; motives
14N10 Enumerative problems (combinatorial problems) in algebraic geometry

Citations:

Zbl 0635.00006