Let $Y$ be a smooth projective variety and $C\subset Y$ a smooth irreducible curve. Let $C_i$ be $C$ with some $i$-fold structure (no need to specify which). Suppose we are in the situation: $$C_{i-1}\subset C_i\subset Y.$$ This just means that $Y$ contains an $i$-fold $C$, containing in turn an $(i-1)$-fold $C$. There is, then, an exact sequence of ideals $$0\to \mathscr I_{i}\to\mathscr I_{i-1}\to\mathscr I_{i-1}/\mathscr I_{i}\to 0,$$ where $\mathscr I_{i}\subset\mathscr O_Y$ is the ideal sheaf of $C_i$.
Question. What does the quotient $\mathscr I_{i-1}/\mathscr I_{i}$ look like (as a sheaf on $Y$)?
I have the feeling it should equal $\mathscr O_C$, but I cannot prove it. Thanks for any help!
No, $\mathscr I_{i-1}/\mathscr I_{i}$ need not be isomorphic to $\mathcal O_C$ .
For $i=2$ we may choose $\mathscr I_1=\mathscr I_C$ and $\mathscr I_2=\mathscr I^2_C$, so that $\mathscr I_1/\mathscr I_2=\mathscr I_C/\mathscr I_C^2$ which is the conormal bundle $N^*_{C/Y}$ to $C$ in $Y$.
This bundle is not in general the trivial bundle $\mathcal O_C$: for example if $C$ is a line in $\mathbb P^2$ its conormal bundle is $\mathcal O_C(-1)$, which has no non-zero section and is thus non-trivial.