Degree and arithmetic genus for projective curves

[sagetrac-gjorgenson]

Implement a function to compute the degree of a projective curve, and generalize the existing arithmetic genus computation for projective plane curves to work for arbitrary projective curves.

CC: @bhutz @miguelmarco

Component: algebraic geometry

Keywords: gsoc2016

Author: Grayson Jorgenson

Branch/Commit: 3427412

Reviewer: Ben Hutz

Issue created by migration from https://trac.sagemath.org/ticket/20848

[sagetrac-gjorgenson]

Branch: u/gjorgenson/ticket/20848

[sagetrac-git]

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

0f96211

20848: first implementation attempt.

[sagetrac-git]

Commit: 0f96211

[miguelmarco]

comment:4

In the case of plane curves, which genus do we want to assignt to non-reduced curves?

[sagetrac-gjorgenson]

comment:5

Oops, I was under the impression that these definitions agreed for arbitrary plane curves. Is the Hilbert polynomial definition still okay for reducible/non-reduced curves? Would it be better to restrict the genus computations to just irreducible curves?

[bhutz]

comment:6

I think genus should be restricted to irreducible. The problem is not the difference between plane and not plane curves. You could have a curve with multiple components of different genus

eg: (y-x)(y^2-x2-x-1)

[sagetrac-gjorgenson]

Changed commit from 0f96211 to 07e6b72

[sagetrac-gjorgenson]

comment:7

I added an is_irreducible check which checks irreducibility by just counting irreducible components.

The new commit shows up in the log for this ticket but no notification seems to have been generated (I think because of the trac server change).

EDIT: actually the new commit shows up as part of this comment now

---

New commits:

07e6b72

20848: added is_irreducible check

[bhutz]

comment:8

• Just checking the number of components for is_irreducible is not good enough. You're better off checking if the defining ideal is prime.

P.<x,y>=ProjectiveSpace(QQ,1)
X=P.subscheme([x^2])
X.is_irreducible(),X.defining_ideal().is_prime()

• since the degree is just the dim! times the leading coefficient of HP, why not do it for projective subschemes instead of just curves?

• Adding an actual REF for Hartshorne would be nice.

[bhutz]

Reviewer: Ben Hutz

[sagetrac-gjorgenson]

Changed commit from 07e6b72 to f4709f5

[sagetrac-gjorgenson]

comment:9

Thanks, made the changes.

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New commits:

f4709f5

20848: generalized degree computation, and improved is_irreducible

[vbraun]

comment:11

Merge conflict

[sagetrac-git]

Changed commit from f4709f5 to df1faa6

[sagetrac-git]

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

df1faa6

20848: merge with 7.3 beta5.

[vbraun]

comment:15

Merge conflict #20839

[sagetrac-git]

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

f775a0f	20839: first implementation attempt.
e0188ce	20839: some changes from review
6564724	20839: merge with ticket 20774
09eea02	20839: some remaining changes from review
4b9ab0a	20839: implemented Serre intersection multiplicity for affine/projective subschemes
cae16fe	20839: improved is_transverse
3427412	20848: merge with ticket 20839

[sagetrac-git]

Changed commit from df1faa6 to 3427412

[sagetrac-gjorgenson]

comment:17

Okay I think the conflict with #20839 is fixed now.

[bhutz]

comment:18

still looks fine for me.

[vbraun]

Changed branch from u/gjorgenson/ticket/20848 to 3427412