# Hilbert scheme of 2 points on an elliptic curve

The Hilbert scheme of 2 points on an elliptic curve $C$, $Hilb^2(C)$, has a natural structure of ruled surface, given by the map $f:Hilb^2(C) \to C$ such that $f(P,Q)=P+Q$.

What can we say about the corresponding locally free sheaf of rank 2 on $C$?

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In this case the corresponding locally free sheaf or rank $2$ on $C$ is the unique indecomposable one.
Dimitri means that it is (up to translations) the unique nontrivial extension of degree 1 $1 \to \mathcal{O}_C→E→\mathcal{O}_C(o) \to 1$, where $o \in C$ is the origin in the group law. –  Francesco Polizzi Nov 5 '12 at 12:35
The curves of the form $D_P=\{x+P \, | \, x \in C \}$ are a $1$-dimensional algebraic family of sections with self-intersection $1$, and moreover one cannot find two disjoint sections... –  Francesco Polizzi Nov 5 '12 at 16:51