next | previous | forward | backward | up | index | toc

# rationalSurface -- compute the ideal I of the rational surface

## Synopsis

• Usage:
I=rationalSurface(P2,d,L,Pn)
I=rationalSurface(P2,d,L,x)
I=rationalSurface(points,d,L,x)
• Inputs:
• P2, a ring, homogeneous coordinate ring of P2,
• points, a list, list of ideals of points in P2
• d, an integer, degree of the desired forms
• L, a list, {r_1,...,r_s} of multiplicities
• Pn, a ring, coordinate ring of Pn
• x, , variable name
• Outputs:
• I, an ideal, of a rational surface

## Description

The function chooses randomly s point p_i in P2 and computes the linear system H=L(d;L) of form of degree d which have multiplicity r_i at the point p_i.

Then the ideal of the image of P2 of under the rational map defined by the linear system H is computed.

 i1 : d=6 o1 = 6 i2 : L=toList(6:2)|toList(2:1) o2 = {2, 2, 2, 2, 2, 2, 1, 1} o2 : List i3 : n=expectedDimension(d,L)-1 o3 = 7 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 : y=symbol y o7 = y o7 : Symbol i8 : betti(I=rationalSurface(P2,d,L,y)) 0 1 o8 = total: 1 11 0: 1 . 1: . 11 o8 : BettiTally i9 : x = symbol x o9 = x o9 : Symbol i10 : Pn=kk[x_0..x_n] o10 = Pn o10 : PolynomialRing i11 : betti(I=rationalSurface(P2,d,L,Pn)) 0 1 o11 = total: 1 11 0: 1 . 1: . 11 o11 : BettiTally i12 : degree I, genus I, dim I o12 = (10, 0, 3) o12 : Sequence i13 : minimalBetti I 0 1 2 3 4 5 o13 = total: 1 11 24 25 15 4 0: 1 . . . . . 1: . 11 20 9 . . 2: . . 4 16 15 4 o13 : BettiTally i14 : d^2-sum(L,r->r^2)== degree I o14 = true i15 : (numList,adjList,ptsList,J)=adjunctionProcess(I); i16 : numList o16 = {(7, 10, 4), 2, (3, 3, 1)} o16 : List i17 : minimalBetti J 0 1 o17 = total: 1 1 0: 1 . 1: . . 2: . 1 o17 : BettiTally

## Ways to use rationalSurface :

• rationalSurface(List,ZZ,List,Symbol)
• rationalSurface(Ring,ZZ,List,Ring)
• rationalSurface(Ring,ZZ,List,Symbol)

## For the programmer

The object rationalSurface is .