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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:
• Outputs:

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 .