Celebratio Mathematica — Thomas

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Works connected to Robin Cope Hartshorne

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R. Hartshorne, E. Rees, and E. Thomas: “Nonsmoothing of algebraic cycles on Grassmann varieties,” Bull. Am. Math. Soc. 80 : 5 (September 1974), pp. 847–851. MR 0357402 Zbl 0289.14011 article

By a cycle \( Z \) of dimension \( r \) on a nonsingular algebraic variety \( X \), we mean a formal linear combination \( Z = \sum n_iY_i \) of irreducible subvarieties \( Y_i \) of dimension \( r \), with integer coefficients \( n_i \). The smoothing problem for cycles asks whether a given cycle \( Z \) is equivalent (for a suitable equivalence relation of cycles, such as rational equivalence or algebraic equivalence) to a cycle \( Z^{\prime} = \sum n^{\prime}_iY^{\prime}_i \), where the subvarieties \( Y^{\prime}_i \) are all nonsingular. Let \( X \) be a nonsingular projective variety of dimension \( n \) over \( \mathbf{C} \). Then for each cycle \( Z \) of dimension \( r \) on \( X \) we can assign a cohomology class \[ \delta(Z)\in H^{2n-2r}(X,\mathbf{Z}) .\] We say that two cycles \( Z \), \( Z^{\prime} \) are homologically equivalent if \( \delta(Z) = \delta(Z^{\prime}) \). Our main result is that there are cycles on certain Grassmann varieties which cannot be smoothed, even for homological equivalence, which is weaker than rational or algebraic equivalence.

Elmer Rees	Related
Robin Cope Hartshorne	Related