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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.

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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 $g$).

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  • $\begingroup$ Thanks! I feel a little foolish to have not realized (2)... $\endgroup$
    – Robin
    Aug 29 '14 at 20:20
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    $\begingroup$ You write "$g=1,\ldots,5,9,10,12,16,\ldots$", but $g=2$ does not occur (while $g=0$ of course does). $\endgroup$ Aug 30 '14 at 4:26
  • $\begingroup$ Yes. Sorry I went too fast! $\endgroup$
    – abx
    Aug 30 '14 at 5:59
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    $\begingroup$ There is now an entry in the OEIS giving the numbers up to a certain cutoff, see oeis.org/A266322. $\endgroup$
    – pbelmans
    Feb 9 '16 at 18:45
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    $\begingroup$ 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. $\endgroup$
    – roy smith
    Sep 5 '17 at 18:40

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