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?
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).