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

Bruguières, A.

Compos. Math. 63, 15-40 (1987).

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

Keywords:

unidecomposable sheaf; morphisms from an elliptic curve to a Grassmann variety

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