Projective Squares in â„™² and Bott‘s Localization Formula

Abstract

We give an explicit description of the Hilbert scheme that parametrizes the closed 0-dimensional subschemes of degree 4 in the projective plane that allows us to afford a natural embedding in a product of Grassmann varieties. We also use this description to explain how to apply Bott‘s localization formula (introduced in 1967 in Bott‘s work [2]) to give an answer for an enumerative question as used by the first time by Ellingsrud and Strømme in [8] to compute the number of twisted cubics on a general Calabi-Yau threefold which is a complete intersection in some projective space and used later by Kontsevich in [16] to count rational plane curves of degree d passing through 3d − 1 points in general position in the plane.