Suppose $M$ is a line bundle on a smooth projective curve $C$ and $N$ is a vector bundle on $C$. Suppose the degree of $M$ is sufficiently big so that ${\rm Hom}(M, N)=0$ and ${\rm Ext}^1(N, M)=0$. Suppose $x$ is a point of $C$. Question: what are extensions of the form $0 \to M \oplus N \to X \to {\mathcal O}_x \to 0$ parametrized by, where $X$ has the form $M_1 \oplus N$ such that $M_1$ is a line bundle of degree larger by $1$ than the degree of $M$? Are such extensions parametrized by ${\rm Ext}^1({\mathcal O}_x, M)$ or by something else?
1 Answer
Since in the map $M\oplus N\to M_1\oplus N$, one has no nonzero maps from $M\to N$, one must have an exact sequence $0\to M\to M_1\to O_x\to 0$. The rest should be easy.
This is just an elaboration of the above asked by the OP. Given an exact sequence $0\to M\oplus N\to M_1\oplus N\to O_x\to 0$ as in the question, we see that the left map $M\to M_1\oplus N$ is given by $(i,0)$ and $N\to M_1\oplus N$ by $(\alpha,\beta)$. Then it is clear that $\beta$ must be an isomorphism. Then, take the isomorphism $M_1\oplus N\to M_1\oplus N$, by $(Id_{M_1}, \beta^{1}) $ and then one sees that the original sequence is equivalent to one given by $M\to M_1\oplus N$ by $(i,0)$ and $N\to M_1\oplus N$ by $(\alpha, Id_N)$, (possibly for a different $\alpha)$. Now, one can take the isomorphism $M_1\oplus N$ to itself by $M_1\to M_1\oplus N$, given by $(Id_{M_1}, 0)$ and $N\to M_1\oplus N$ by $(\alpha, Id_N)$ and then you get an equivalent extension where $M\to M_1\oplus N$ is $(i,0)$ and $N\to M_1\oplus N$ is $(0,Id_N)$ and the rest should be clear.

$\begingroup$ Thank you very much for offering to help! I am sorry, I understand that there is an exact sequence $0 \to M \to M_1 \to O_x \to 0$ but I don't think this means that my extensions are parametrized by $Ext^1(O_x, M)$. Indeed, what if the induced map $N \to M_1$ from the map $M \oplus N \to X$ in my extension is nonzero? Maybe in that case one could still have a splitting $X=M_1 \oplus N$? Or one couldn't? Could you please elaborate on the rest of the answer? Thank you again! $\endgroup$ Jul 4, 2019 at 20:34

$\begingroup$ Sorry, if you see that my question is not that easy, could you either elaborate on your answer or delete it so that other people can see that there is no answer and be more likely to answer the question? Thank you very much! $\endgroup$ Jul 4, 2019 at 22:35

$\begingroup$ First of all, my answer is correct. Have you tried to work out that from what I said, you can show what you want? Can you show that the extension you have is equivalent to the extension $0\to M\oplus N\to M_1\oplus N\to O_x\to 0$ where the first map is $M\to M_1\oplus N$, given by $(i,0)$ and the second map $N\to M_1\oplus N$ is given by $(0, Id_N)$? $\endgroup$– MohanJul 5, 2019 at 0:35

$\begingroup$ Thank you very much! This is exactly what I don't know how to do. Could you explain it or at least give a hint? It's not a homework problem, I need it for my research paper on another subject. I have been thinking about this question for 3 days. $\endgroup$ Jul 5, 2019 at 1:18