Basic intersection analysis for algebraic curves

[sagetrac-gjorgenson]

Implement basic intersection analysis, such as when given two curves and a point, determine if the point is in the intersection of the two curves, and if so, compute the intersection multiplicity of the curves at that point if defined. Also, given a projective curve, determine if it is a complete intersection.

CC: @bhutz @miguelmarco

Component: algebraic geometry

Keywords: gsoc2016

Author: Grayson Jorgenson

Branch/Commit: cae16fe

Reviewer: Ben Hutz

Issue created by migration from https://trac.sagemath.org/ticket/20839

[sagetrac-gjorgenson]

Branch: u/gjorgenson/ticket/20839

[sagetrac-git]

Commit: f775a0f

[sagetrac-git]

Branch pushed to git repo; I updated commit sha1. New commits:

f775a0f

20839: first implementation attempt.

[bhutz]

Reviewer: Ben Hutz

[bhutz]

comment:4

• intersects_at(): why not but both attempts in the same try block?

• is_complete_intersection: Boolean can go on the same line at OUTPUT
  not the radical ideal per our discussion today

here is another noncomplete intersection example

P.<x,y,z,w>=ProjectiveSpace(QQ,3)
X= Curve([x*z-y^2,z*(y*w-z^2) - w*(x*w-y*z)])
X.is_complete_intersection()

• intersection_multiplicity: an integer can go on the same line as OUTPUT

good these add up to 4 as bezout's theorem implies

K.<i>=QuadraticField(-1)
A.<x,y>=AffineSpace(K,2)
C = Curve([x^2-y])
D = Curve([x^2+y^2])
for t in C.intersection(D).rational_points():
    C.intersection_multiplicity(D,t)

also good

K.<i>=QuadraticField(-1)
A.<x,y>=AffineSpace(K,2)
C = Curve([y^2-x^3])
D = Curve([(x^2+y^2)^2 - 4*x^2*y^2])
for t in C.intersection(D).rational_points():
    C.intersection_multiplicity(D,t)

so for plane curves this looks ok. Does this really work in dimension greater than 2? I don't think it does, I think the intersection number possibly has lower order terms. Regardless, in looking at this, the following example died

K.<i>=QuadraticField(-1)
A.<x,y,z>=AffineSpace(K,3)
C = Curve([x^2-y,z^2+x^2])
D = Curve([x^2+y^2,x^2+z])
for t in C.intersection(D).rational_points():
    C.intersection_multiplicity(D,p)

• I think an transverse check would be nice too:

is_transverse() - Returns true if and only if the point p is a nonsingular point of both plane curves C and D and the curves have distinct tangents there.

[sagetrac-git]

Changed commit from f775a0f to 09eea02

[sagetrac-git]

Branch pushed to git repo; I updated commit sha1. Last 10 new commits:

bbffa39	20774: Merge this ticket with 20697.
4fce7ed	20774: Merge this ticket with 20676.
4655c72	20774: Implement multiplicity, tangents, and is_ordinary_singularity
e1f1529	20774: Merge with 7.3 beta3
e6f1a93	20774: Changes from first review.
1f0d7ab	20774: Merge with 7.3 beta4.
bfcd05e	20774: Added some error tests, and updated error messages.
4282a4f	20774: addressed small bug with point search
6564724	20839: merge with ticket 20774
09eea02	20839: some remaining changes from review

[sagetrac-gjorgenson]

comment:6

Thanks, I made most of the changes, but haven't yet addressed the intersection_multiplicity issue. Is there a good way to generalize the intersection multiplicity computations to work for space curves, or would it be best to make them specific to plane curves?

I really don't have much homological algebra background, but if I understand correctly, I think a general definition of intersection multiplicity is given by Serre's Tor formula. I think Singular has some functionality for working with the needed constructions, but I'm not sure whether it's enough to be able to do multiplicity computations.

[sagetrac-gjorgenson]

comment:7

New commit didn't automatically appear.

---

New commits:

4b9ab0a

20839: implemented Serre intersection multiplicity for affine/projective subschemes

[sagetrac-gjorgenson]

Changed commit from 09eea02 to 4b9ab0a

[bhutz]

comment:8

This looks fine except that is_transverse should return False when one of the points is singular.

[sagetrac-gjorgenson]

Changed commit from 4b9ab0a to cae16fe

[sagetrac-gjorgenson]

New commits:

cae16fe

20839: improved is_transverse

[vbraun]

Changed branch from u/gjorgenson/ticket/20839 to cae16fe