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.
@article{1437,
title = {Projective Squares in P2 and Bott's Localization Formula},
journal = {CUBO, A Mathematical Journal},
volume = {12},
year = {2010},
language = {en},
url = {http://dml.mathdoc.fr/item/1437}
}
Rojas, Jacqueline; Mendoza, Ramon; da Silva, Eben. Projective Squares in ℙ² and Bott’s Localization Formula. CUBO, A Mathematical Journal, Tome 12 (2010) . http://gdmltest.u-ga.fr/item/1437/