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

ΔA, 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

MH, complement of hyperplane

arrangement, 63

m(ω, A), minimum value of ω on A, 33

MV(K1, . . . , 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