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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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    Yes, I was. I actuall$y$ meant to ask him this question after lecture last time but I had to make it to another class.2012-05-05

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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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    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.2012-05-06