expectedDimension -- compute the expected dimension of a linear system on a rational surface

Synopsis

Usage:

expectedDimension(d,L)
Inputs:
- d, an integer, degree of the desired forms
- L, a list, {r_1,...,r_s} of multiplicities
Outputs:
- an integer, the expected dimension of L(d;r_1p_1,...,r_sp_s)

Description

computes the expected dimension max(0,binomial(d+2,2)-sum binomial(r_i+1,2)) of a linear system with assigned base points on P2

i1 : d=8

o1 = 8

i2 : L=toList(3:3)|toList(2:4)|{1}

o2 = {3, 3, 3, 4, 4, 1}

o2 : List

i3 : expectedDimension(d,L)

o3 = 6

i4 : kk=ZZ/nextPrime(10^3)

o4 = kk

o4 : QuotientRing

i5 : t=symbol t

o5 = t

o5 : Symbol

i6 : P2=kk[t_0..t_2]

o6 = P2

o6 : PolynomialRing

i7 : betti(H=linearSystemOnRationalSurface(P2,d,L))

            0 1
o7 = total: 1 6
         0: 1 .
         1: . .
         2: . .
         3: . .
         4: . .
         5: . .
         6: . .
         7: . 6

o7 : BettiTally

Ways to use expectedDimension :

expectedDimension(ZZ,List)

For the programmer

The object expectedDimension is a method function.