## The scheme of morphisms from an elliptic curve to a Grassmannian.(English)Zbl 0664.14005

The author studies the scheme $$M_ d(X;p,n)$$ parametrizing morphisms of degree $$d$$ from an elliptic curve X to a Grassmann variety Gr(p,n). He proves that $$M_ d(X;p,n)$$ is smooth if and only if $$p=1$$, or $$p=n-1$$, or $$d=2$$; it is irreducible of dimension nd if $$d\geq n$$; if $$d>n$$, it is irreducible if and only if it is smooth. More precisely, if $$d>n$$, the author shows that the irreducible components are the Zariski closures of certain locally closed subschemes $$M_ d^{a,b}(X;p,n)$$, for $$0\leq a<p$$, $$0\leq b<n-p$$ and $$a+b=n-d$$, of dimension $$nd+ab$$. These subschemes are defined as follows: If $$f\in M_ d(X;p,n)$$ is a closed point, let $$0\to V_ f\to E_ X\to Q_ f\to 0$$ denote the pullback of the universal sequence on Gr(p,n). Then set $$M_ d^{a,b}(X;p,n)=\{f;\quad h^ 0(V_ f)=a,\quad h^ 0(Q_ f)=b\}$$.
The last part of the paper gives sufficient conditions for three sheaves on X to fit into an exact sequence. If the sheaves are locally free and the sheaf of highest rank is trivial, this is related to the existence of maps from X into a corresponding Grassmann variety, with fixed pullbacks of the trivial bundles.
Reviewer: R.Piene

### MSC:

 14D20 Algebraic moduli problems, moduli of vector bundles 14F05 Sheaves, derived categories of sheaves, etc. (MSC2010) 14H45 Special algebraic curves and curves of low genus 14H52 Elliptic curves
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### References:

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