Schubert Calculus for Complete Quadrics

SL2-Regular Subvarieties of Complete Quadrics

“…1. [26] X can be obtained by the following sequence of blowups: in the naive compactification P n−1 of X 0 , first blow up the locus of rank 1 quadrics; then blow up the strict transform of the rank 2 quadrics; . .…”

Polynomial vector fields with algebraic trajectories

“…Remark. The variety B is the well known variety of complete conics (see [9], [11]). It is constructed to solve the indeterminacies of the map…”

“…Recall that the exceptional divisor of the blowup is E := P(N ) where N = N V P 5 stands for the normal bundle. We have an explicit description for N see [11,Proposition 4.4.]. Notation as in (4, p. 2) with k = 1, n = 2, we have N V P 5 = OP2(2) ⊗ Sym 2 (T ∨ ).…”

“…Using the varieties of complete quadrics of any dimension in P n (see [11]) it is possible to find a compactification of the space of dimension one foliations in P n that leave invariant a smooth quadric.…”

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“…In the case of quadratic Veronese embeddings where there is a unique plane quadric through any two points, it is not dif¢cult to modify the results of the previous sections to achieve similar results. We do not do this, however, as this case is already understood from the point of view of complete quadrics [29,30].& EXAMPLE 4.14. It is interesting to examine Theorem 4.12 in cases where r 2 X is well understood.…”