I'm having some questions about the geometry of the cubic 3-fold. Every variety is over $\mathbb C$.
Take $Y$ a smooth cubic 3-fold in $\mathbb P^4$ and $E$ a curve of degree 6 and genus 1 contained in it. First assume it is smooth.
If i study the exact sequence $$0\to I_E(2)\to O_Y(2) \to j_*O_E(2)\to 0$$ I arrive at the conclusion that $H^0(Y,I_E(2))$ has dimension 3, and $E$ is contained in the zero locus of a section of the linear system $O_Y(2)$ that is smooth. I can compute its tangent bundle using the adjunction formula and as it is a complete intersection, I can prove that its $h^{0,1}$ is zero, and therefore it is a smooth K3 surface, let us call it $S$.
Now, I'd like to prove that $O_S(E)$ is generated by 2 sections. I belive I can compute the dimension of $H^0(S,O_S(E))$ using Riemann-Roch, but how do I get that the sections actually generate $O_S(E)$.
Thanks.