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We learned the following relationship between the degree and genus of plane curves in my algebraic geometry course:

\begin{array} a \text{degree} &d &1 &2 &3 &4 &5 &6 &7 & \dots\\ \text{genus} &g &0 &0 &1 &3 &6 &10 &15 & \dots \end{array}

So there are no plane curves of genus 2, 4, 5, etc. My question is: what is the relationship between degree and genus for space curves? In particular, do there also exists gaps like this? Why or why not?

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I don't think that the tabular command works in this website as you can see in this meta thread. –  Adrián Barquero May 5 '12 at 22:08
I tried to arrange your table with an array, but I didn't want to delete the whole thing just yet in case I made a mistake. –  Adrián Barquero May 5 '12 at 22:10
Thanks for fixing it. I'll delete that ugly unrendered stuff now. –  Derek Allums May 5 '12 at 22:13
By the way unit3000-21, I have the same table in my algebraic geometry notes from this week. Were you by any chance taking Frank's class? (If you were then you'll know which Frank, if not I'll just delete my comment, I'm just curious). Since the table looked familiar, that's one reason I decided to edit it ;) –  Adrián Barquero May 5 '12 at 22:25
Yes, I was. I actually meant to ask him this question after lecture last time but I had to make it to another class. –  Derek Allums May 5 '12 at 22:33

1 Answer 1

up vote 4 down vote accepted

In space, the genus is not determined completely by the degree. This paper by Harris mentions some known bounds, and this thesis seems to have some relevant results (see Chapter 2).

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I'll take a look at the paper tomorrow when I'm on campus. The thesis has some nice results (e.g. Theorem 2.1.1), but I think your first sentence is really what answers my question. In space, the degree does not completely determine the genus and so "are there gaps..." is sort of a meaningless question, right? –  Derek Allums May 6 '12 at 14:30
You can still ask about gaps in the sense of which pairs $(d,g)$ are possible in space. –  Ted May 6 '12 at 19:10
Right: but what I meant was that, by your answer, the solution won't be as simple as constructing a table and observing which integers don't appear as we go up in degree. However, the question "which pairs $(d,g)$ are possible in space?" is actually more interesting. –  Derek Allums May 6 '12 at 20:17

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