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# linearSystemOnRationalSurface -- compute a linear system on a rational surface

## Synopsis

• Usage:
H=linearSystemOnRationalSurface(P2,d,L)
H=linearSystemOnRationalSurface(points,d,L)
• Inputs:
• P2, a ring, homogeneous coordinate ring of P2
• points, a list, a list of ideals defining points in P2
• d, an integer, degree of the desired forms
• L, a list, {r_1,...,r_s} of multiplicities
• Outputs:
• H, , a 1xn matrix whose entries are a bases of L(d;r_1p_1,...,r_sp_s)

## Description

The function chooses randomly s point p_i in P2 and computes the linear system of form of degree d which have multiplicity r_i at the point p_i

 i1 : kk=ZZ/nextPrime(10^3) o1 = kk o1 : QuotientRing i2 : t=symbol t o2 = t o2 : Symbol i3 : P2=kk[t_0..t_2] o3 = P2 o3 : PolynomialRing i4 : d=8 o4 = 8 i5 : L=toList(3:3)|toList(2:4)|{1} o5 = {3, 3, 3, 4, 4, 1} o5 : List i6 : expectedDimension(d,L) o6 = 6 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 linearSystemOnRationalSurface :

• linearSystemOnRationalSurface(List,ZZ,List)
• linearSystemOnRationalSurface(Ring,ZZ,List)

## For the programmer

The object linearSystemOnRationalSurface is .