baseLocus -- compute the locus where a graded module (or O(D) of a Weil divisor) is not globally generated

Synopsis

Usage:

I = baseLocus(D)

I = baseLocus(M)
Inputs:
- D, an instance of the type WeilDivisor,
- M, a module,
Outputs:
- I, an ideal,

Description

Given a graded module $M$ with degree 0 global sections $s1, ..., sd$, this computes the locus where the $si$ do not generate $M$. Given a Weil divisor $D$, this computes the base locus of $O(D)$. For example, consider the rulings on $P^1 \times P^1$.

i1 : R = QQ[x,y,u,v]/ideal(x*y-u*v);

i2 : D = divisor( ideal(x,u) )

o2 = Div(x, u)

o2 : WeilDivisor on R

i3 : baseLocus(D)

o3 = ideal 1

o3 : Ideal of R

Next we consider an example of a point on an elliptic curve.

i4 : R = QQ[x,y,z]/ideal(y^2*z-x*(x+z)*(x-z));

i5 : D = divisor(ideal(y, x))

o5 = Div(y, x)

o5 : WeilDivisor on R

i6 : baseLocus(D)

o6 = ideal (y, x)

o6 : Ideal of R

i7 : baseLocus(2*D)

o7 = ideal 1

o7 : Ideal of R

Ways to use baseLocus :

baseLocus(Module)
baseLocus(WeilDivisor)

For the programmer

The object baseLocus is a method function.