13

I have two related questions about smooth complete algebraic curves (over $\mathbb{C}$).

  1. Does there exist a smooth complete algebraic curve $X$ that cannot be embedded as a complete intersection in $\mathbb{P}^n$ for any $n$ (nb: this is different from asking if every smooth curve in $\mathbb{P}^n$ is a complete intersection, which is of course false; e.g. the twisted cubic)? I expect that the answer is "yes", though it might be "no" for specific genera (and I'd be interested in known these genera). If you bounded $n$, then probably you could use the fact that the moduli space of curves is of general type for large genus to prove this.

  2. Fix a genus $g$. Does there exist some $n$ such that $\mathbb{P}^n$ contains a smooth genus $g$ curve as a complete intersection? I'm not really sure if the answer should be yes or no; if it is no, then I'd be interested in knowing which $g$ satisfy this.

Robin
  • 141

1 Answers1

25

1) The genus of a complete intersection of multidegree $(d_1,\ldots ,d_{n-1})$ in $\mathbb{P}^n$ is $g=1+\frac{1}{2} d_1\cdot \ldots \cdot d_{n-1}(\sum d_i-n-1)$ (just compute the degree of the canonical bundle). This gives very particular values for $g$: $0, 1, 3, 4, 5, 6, 9, 10, 13, 15, 16,\dotsc $ . Any curve whose genus is not in this list cannot be realized as a complete intersection.

2) Even if $g$ is in that list, for $g>5$ a general curve of genus $g$ cannot be realized as a complete intersection, since the number of moduli of such complete intersection is smaller than $3g-3$ (the number of moduli of a general curve of genus $g$).

Zach Teitler
  • 6,197
abx
  • 37,143
  • 3
  • 82
  • 142
  • Thanks! I feel a little foolish to have not realized (2)... – Robin Aug 29 '14 at 20:20
  • 4
    You write "$g=1,\ldots,5,9,10,12,16,\ldots$", but $g=2$ does not occur (while $g=0$ of course does). – Noam D. Elkies Aug 30 '14 at 04:26
  • Yes. Sorry I went too fast! – abx Aug 30 '14 at 05:59
  • Shouldn't 6 also be in that list, using a planar curve of degree 5? I also think 12 should not be in the list, whilst 13 should (using (3,2,2)) and so does 15 (as a curve of degree 7). – pbelmans Dec 27 '15 at 13:41
  • You are absolutely right. Sorry I didn't pay too much attention to the precise numbers -- I just wanted to indicate the principle. – abx Dec 27 '15 at 17:22
  • 4
    There is now an entry in the OEIS giving the numbers up to a certain cutoff, see https://oeis.org/A266322. – pbelmans Feb 09 '16 at 18:45
  • 1
    The answer of abx of course also reveals that the canonical bundle of a c.i. curve of genus ≥ 2 is very ample, hence no hyperelliptic curve of any genus ≥ 2 can occur as a c.i. This is just to illustrate explicitly the remark of abx that even among the allowable genera, exceptions exist. – roy smith Sep 05 '17 at 18:40
  • Does the above-given equation for the genus hold only for polynomials that do not have singularity? – Alm Jul 23 '18 at 14:53
  • No, it is still valid for a singular curve $X$ if you have in mind the arithmetic genus, that is $\dim H^1(X,\mathcal{O}_X)$. For singular curves there is also the geometric genus, that is, the genus of the normalization of $X$; it is equal to the arithmetic genus minus the contribution of the singularities. – abx Jul 24 '18 at 04:30
  • @Eoin: No, the computation of the canonical bundle does not necessitate any assumption on the hypersurfaces. – abx Jul 27 '19 at 04:27