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Density of elliptic solitons. (English) Zbl 0810.14011
Let \((X,q)\) be an elliptic curve. An (elliptic) tangential cover is a finite pointed morphism \(\pi : (\Gamma, p) \to (X,q)\), where \(\Gamma\) is a (projective) curve and \(p\) is a smooth point, such that the induced map on the Jacobians \(\pi^* : \text{Jac} X \cong X \to \text{Jac} \Gamma\) is tangent to the image of \(\Gamma\) under the Abel map based at the point \(p\).
The theorem proved in the paper under review is the following. The smooth curves which are (elliptic) tangential covers of genus \(g\) are a dense subset with respect to the complex topology in the moduli space \({\mathcal M}_ g\) of all curves of genus \(g\). The same holds for hyperelliptic tangential covers and the moduli space of hyperelliptic curves. – To prove the theorem the fact is used, that there is a rank 2 vector bundle \(E\) on the elliptic curve \(X\) such that every tangential cover factors through \(\Gamma \to S = \text{Proj} (E) \to X\). The cover is minimal if the first map is an embedding. Now deformation theoretical arguments are used to prove the theorem.
Applications and questions are discussed in a last section. In particular, such tangential covers correspond to certain solutions of the KP equation which have doubly periodic behaviour (elliptic solitons). Using the theorem above the authors show that these elliptic solitons considered as subspace of the Segal-Wilson Grassmannian are dense in the whole Grassmannian.

MSC:
14H10 Families, moduli of curves (algebraic)
14F10 Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials
35Q51 Soliton equations
35Q53 KdV equations (Korteweg-de Vries equations)
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