Draft talk:Complete algebraic curve

Should this be projective curve or part of complete variety?

@TakuyaMurata: At the current stage, should this be separate from complete variety? There isn't that much in the article on general complete varieties right now, so this could certainly be added / merged into that article. Alternatively, this could be turned into the proposed article on projective curve since the material here is essentially focused on the projective case. — MarkH21 (talk) 19:07, 21 April 2019 (UTC)[reply]

No, I don’t think complete variety should have too much stuff on curves. Yes that article is short but that’s ok and it would look very out-of-place to have the materials of this article there. Also, this article is supposed to be the same as projective curve; as “projective” and “complete” can be used interchangeably for curves. When this article is moved to mainspace, projective curve will be redirected to this article. —- Taku (talk) 11:03, 22 April 2019 (UTC)[reply]

Either “complete curve” or “projective curve” works as the article title but it seems the former is more common and also is better especially when one talks about a family of curves (in that case, the distinction between “complete (i.e., proper)” and “projective” matters). —- Taku (talk) 11:53, 22 April 2019 (UTC)[reply]

Right, my point was that my impression is that "projective curve" is more commonly preferred over "complete curve". For instance: Hartshorne's book, Qing Liu's book, Vakil's notes, and Arbarello–Cornalba–Griffiths–Harris exclusively use "projective curve" (except when Hartshorne shows that complete implies projective for curves). — MarkH21 (talk) 21:12, 23 April 2019 (UTC)[reply]

Stuff from Geometry of an algebraic curve

I’m not sure where the following stuff should go; so I’m them here temporarily. —- Taku (talk) 11:21, 22 April 2019 (UTC)[reply]

Joe Harris and Ian Morrison. Moduli of curves.
Kollár, János, "Chapter 1", Book on Moduli of Surfaces

http://arxiv.org/pdf/1503.04465v2.pdf

In the context of degeneration/deformation of curves, it is imperative to talk about "reducible" curves with singularities.

“Flat families of curves” By which we mean

\pi :{\mathcal {X}}\to S

such that

$\pi$ is flat (but need not be proper)
$\pi ^{-1}(t)$ is a smooth curve for all t ≠ 0.

By the degeneration or specialization as t → 0 we mean

Any complete or projective curve can be embedded into ${\mathbb {P} }^{3}$

I am not sure that this is correct for singular curves, and I believe that there exist curve singularities in ${\mathbb {P} }^{n}$ that are not isomorphic to curve singularities in ${\mathbb {P} }^{n-1},$ for $n>3.$ D.Lazard (talk) 08:40, 26 April 2019 (UTC)[reply]

Please note I haven't checked the accuracy yet. Yes, perhaps you're right; the argument in Hartshorne does not seem to use smoothness but might rely on smoothness somehow. In any case, we don't need nor should claim something not backed by sources (and so I will fix that). -- Taku (talk) 11:47, 26 April 2019 (UTC)[reply]

Should this be projective curve or part of complete variety?

Stuff from Geometry of an algebraic curve

Any complete or projective curve can be embedded into P 3 {\displaystyle {\mathbb {P} }^{3}}

Any complete or projective curve can be embedded into ${\mathbb {P} }^{3}$