Index of Notation #m,p, Schubert number, 5 − →, rational map, 79 , cover in a poset, 91, 126 A, support of a polynomial, 3, 26 A+, lift of A, 30 α(κ), sequence associated to a subset κ ⊂ [m], 174 β(κ), sequence associated to a subset κ ⊂ [m], 175 ( [m+p] p ) , Bruhat order, 97 , Schubert condition (m, m+2,. . . , m+p), 117, 122 C, complex numbers, 1 C∗, nonzero complex numbers, 1 Cm,p, product of chains, 96 C{t}, field of Puiseaux series, 34 C[X], homogeneous coordinate ring of projective variety X, 30 d(A), number of solutions to system with support A, 26 d(α), degree of Schubert variety Xα, 128 deg(X), degree of subvariety X, 29 Δ A , convex hull of A, 3, 26 Δω, regular polyhedral subdivision induced by ω, 42 Δp, positive chamber of hyperplane complement, 65 des(w), descent of Grassmannian condition w, 183 d(f1, . . . , fn), number of solutions to f1 = · · · = fn = 0, 26 eα, basis element of ∧pCm+p, 96 E•(s), flag of polynomials vanishing to different orders at s, 131 F•, flag of subspaces, 117 F•(t), osculating flag, 119, 121 F (a m), flag manifold, 180 G, linear algebraic group, 179 g+, lift of map g to sphere, 80 G/P , flag manifold, 179 γ, rational normal curve, 6, 118, 121 Gr(m, Cm+p−1[t]), Grassmannian, 5 Gr(p, m+p), Grassmannian, 96, 116 H, hyperplane arrangement, 63 HX(d), Hilbert function of projective variety X, 31 hX(d), Hilbert polynomial, 30 Im,p, Pl¨ ucker ideal, 98 inω(F ), facial system, 33 inω(f), initial form, 33, 40 inω(XA), initial scheme, 40 Ka, Kostka number, 143 κ = κ1 + · · · + κk, sum of a sequence, 174 λ(P ), number of linear extensions of poset P , 92 LG(m), Lagrangian Grassmannian, 175 lg(w), length of a permutation, 103, 179 mdeg(ρ), mapping degree of ρ, 19 M H , complement of hyperplane arrangement, 63 m(ω, A), minimum value of ω on A, 33 MV(K 1 , . . . , Kn), mixed volume of convex bodies K1, . . . , Kn, 3, 27 N, natural numbers, 1 [n], set of integers {1, 2, . . . , n}, 2 OG(m), orthogonal Grassmannian, 173 OP , order polytope of poset P , 92 PA, projective space with coordinates indexed by A, 27 ϕA, monomial parameterization, 27 Pn, complex projective space, 2 Pω, lifted polytope, 42 ψ, aﬃne-linear map, 63 Q, rational numbers, 1 R, real numbers, 1 R∗, nonzero real numbers, 1 R , positive real numbers, 1, 65 Rn , positive orthant, 49 ρ, rational function, 19 195

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