Mathematics – Algebraic Geometry
Scientific paper
1996-06-10
Variedades Abelianas y Funciones Theta, Ap. Mat. Serie Investigaci\'on No. 13, Sociedad Matem\'atica Mexicana, M\'exico 1998
Mathematics
Algebraic Geometry
Minor changes in Section 5 and References
Scientific paper
The aim of this paper is to offer an algebraic construction of infinite-dimensional Grassmannians and determinant bundles (and therefore valid for arbitrary base fields). As an application we construct the $\tau$-function and formal Baker-Akhiezer functions over arbitrary fields, by proving the existence of a ``formal geometry'' of local curves analogous to the geometry of global algebraic curves. We begin by defining the functor of points, $\fu{\gr}(V,V^+)$, of the Grassmannian of a $k$-vector space $V$ in such a way that its rational points are precisely the points of the Grassmannian defined by Segal-Wilson, although the points over an arbitrary $k$-scheme $S$ have been not previously considered. This definition of the functor $\fu{\gr}(V,V^+)$ allows us to prove that it is representable by a separated $k$-scheme $\gr(V,V^+)$. Using the theory of determinants of Knudsen and Mumford, the determinant bundle is constructed. This is one of the main results of the paper because it implies that we can define ``infinite determinants'' in a completely algebraic way.
Muñoz Porras José María
Plaza Martín Francisco José
Vázquez Álvarez A.
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