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4.iii.b. Quantum Schubert calculus

The space Mqk,n of degree q maps M : P1 --> Gr(k,n) is a smooth quasi-projective variety [Cl]. A point t in P1 and a Schubert variety Ya F. together impose a quantum Schubert condition on maps M in Mqk,n by requiring that

M(t)  lies in Ya F. .

The set of such maps has codimension |a| in the space Mqk,n. The quantum Schubert calculus of enumerative geometry asks the following question.

Question 4.7   Given a1, a2, ..., as in Cn,k with |a1|+|a2|+...+|as| equal to the dimension of Mqk,n (which is qn+k(n-k)), how many maps M in Mqk,n satisfy

M(ti)  lies in Yai F.i for each i = 1, 2, ..., s ,

where t1, t2, ..., ts are general points in P1 and F.1, F.2, ..., F.s are general flags ?

Algorithms to compute this number were proposed by Vafa [Va] and Intriligator [In] and proven by Siebert and Tian [ST] and by Bertram [Ber]. A simple quantum Schubert condition defines a subvariety of codimension 1,

M(t) meets L non-trivially , (4.14)
where L is a (n-k)-plane. Let d(q;k,n) be the number of maps M in Mqk,n satisfying dimMqk,n = qn+k(n-k)-many general simple quantum Schubert conditions (4.14). A combinatorial formula for this number was given by Ravi, Rosenthal, and Wang [RRW]. For a survey of this particular enumerative problem and its importance to linear systems theory, see [So10].

Theorem 4.8 ([So6, Theorem 1.1])   Let q be a non-negative integer and n>k>0 and set N:=dim Mqk,n. Then there exist t1, t2, ..., tN in RP1 and (n-k)-planes L1, L2, ..., LN in Rn such that there are exactly d(q;k,n) maps M in Mqk,n(C) satisfying

M(ti) meets Li non-trivially for each i = 1, 2, ..., N ,

and all of them are real.

As with the classical Schubert calculus, the question of whether the general quantum Schubert calculus is fully real remains open.

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Next: 4.iii.c Schubert calculus of flags
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