E. Gómez González, J. M. Muñoz Porras, F. J. Plaza Martín
The aim of the paper is twofold. First, some results of Shiota and Plaza-Martin on Prym varieties of curves with an involution are generalized to the general case of an arbitrary automorphism of prime order. Second, the equations defining the moduli space of curves with an automorphism of prime order as a subscheme of the Sato Grassmannian are given.
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In this paper we study Prym varieties and their moduli space using the well known techniques of the infinite Grassmannian. There are three main results of this paper: a new definition of the BKP hierarchy over an arbitrary base field (that generalizes the classical one over the complex numbers; a characterization of Prym varieties in terms of dynamical systems, and explicit equations for the moduli space of (certain) Prym varieties. For all of these problems the language of the infinite Grassmannian, in its algebro-geometric version, allows us to deal with these problems from the same point of…
Toward the Universal Mumford form on Sato Grassmannians
本文利用 Virasoro 代数的流在 Sato Grassmannian 乘积上构造局部普适 Mumford 形式，并以 Neveu–Schwarz 流推广得到超 Sato Grassmannian 上的普适超 Mumford 形式。
The degeneration of the hyperelliptic sigma function is studied. We use the Sato Grassmannian for this purpose. A simple decomposition of a rational function gives a decomposition of Plücker coordinates of a frame of the Sato Grassmannian. It then gives a decomposition of the tau function corresponding to the degeneration of a hyperelliptic curve of genus $g$ in terms of the tau functions corresponding to a hyperelliptic curve of genus $g-1$. Since the tau functions are described by sigma functions, we get the corresponding formula for the degenerate hyperelliptic sigma function.
Every Grassmannian, in its Plücker embedding, is defined by quadratic polynomials. We prove a vast, qualitative, generalisation of this fact to what we call Plücker varieties. A Plücker variety is in fact a family of varieties in exterior powers of vector spaces that, like the Grassmannian, is functorial in the vector space and behaves well under duals. A special case of our result says that for each fixed natural number k, the k-th secant variety of any Plücker-embedded Grassmannian is defined in bounded degree independent of the Grassmannian. Our approach is to take the limit of a Plücker va…
An algebraic approach is developed to define and study infinite dimensional grassmannians. Using this approach a quantum deformation is obtained for both the ind-variety union of all finite dimensional grassmannians, and the Sato grassmannian introduced by Sato. They are both quantized as homogeneous spaces, that is together with a coaction of a quantum infinite dimensional group. At the end, an infinite dimensional version of the first theorem of invariant theory is discussed for both the infinite dimensional special linear group and its quantization.
The Super Mumford Form and Sato Grassmannian
本研究推广了Kontsevich等人的经典构造，利用超黎曼曲面模空间与佐藤格拉斯曼流形的关联，证明了特定线丛上平坦全纯联络的存在性及超共形Noether正规化引理。
The construction of families of Sato Grassmannians, their determinant line bundles and the extensions induced by them are given. The base scheme is an arbitrary scheme.
A new relation between Prym varieties of arbitrary morphisms of algebraic curves and integrable systems is discovered. The action of maximal commutative subalgebras of the formal loop algebra of GL(n) defined on certain infinite-dimensional Grassmannians is studied. It is proved that every finite-dimensional orbit of the action of traceless elements of these commutative Lie algebras is isomorphic to the Prym variety associated with a morphism of algebraic curves. Conversely, it is shown that every Prym variety can be realized as a finite-dimensional orbit of the action of traceless diagonal el…
We study moduli spaces of K3 surfaces endowed with a Nikulin involution and their image in the moduli space R_g of Prym curves of genus g. We observe a striking analogy with Mukai's well-known work on ordinary K3 surfaces. Many of Mukai's results have a very precise Prym-Nikulin analogue, for instance: (1) A general Prym curve from R_g is a section of a Nikulin surface if and only if g\leq 7, g\neq 6. (2) R_7 has the structure of a Mori fibre space over the corresponding moduli space of polarized Nikulin surfaces. (3) In the case of genus next to maximal (g=6), the Prym-Nikulin locus in R_6 is…
We study the multiplication and comultiplication in equivariant cohomology of Sato Grassmannian
Algebraic and geometric structures associated with Birkhoff strata of Sato Grassmannian are analyzed. It is shown that each Birkhoff stratum $Σ_S$ contains a subset $W_{\hat{S}}$ of points for which each fiber of the corresponding tautological subbundle $TB_{W_S}$ is closed with respect to multiplication. Algebraically $TB_{W_S}$ is an infinite family of infinite-dimensional commutative associative algebras and geometrically it is an infinite tower of families of algebraic curves. For the big cell the subbundle $TB_{W_\varnothing}$ represents the tower of families of normal rational (Veronese)…
In this paper we study the Hurwitz scheme in terms of the Sato Grassmannian and the algebro-geometric theory of solitons. We will give a characterization, its equations and a show that there is a group of Virasoro type which uniformizes it.
In this paper the moduli space of Higgs pairs over a fixed smooth projective curve with extra formal data is defined and it is endowed with a scheme structure. We introduce a relative version of the Krichever map using a fibration of Sato Grassmannians and show that this map is injective. This fact and the characterization of the points of the image of the Krichever map allow us to prove that this moduli space is a closed subscheme of the product of the moduli of vector bundles (with formal extra data) and a formal anologue of the Hitchin base. This characterization also provide us the method …
Algebraic varieties and curves arising in Birkhoff strata of the Sato Grassmannian Gr${^{(2)}}$ are studied. It is shown that the big cell $Σ_0$ contains the tower of families of the normal rational curves of all odd orders. Strata $Σ_{2n}$, $n=1,2,3,...$ contain hyperelliptic curves of genus $n$ and their coordinate rings. Strata $Σ_{2n+1}$, $n=0,1,2,3,...$ contain $(2m+1,2m+3)-$plane curves for $n=2m,2m-1$ $(m \geq 2)$ and $(3,4)$ and $(3,5)$ curves in $Σ_3$, $Σ_5$ respectively. Curves in the strata $Σ_{2n+1}$ have zero genus.
In this paper, an explicit hierarchy of differential equations for the $τ$-functions defining the moduli space of curves with automorphisms as a subscheme of the Sato Grassmannian is obtained. The Schottky problem for Riemann surfaces with automorphisms consists of characterizing those p.p.a.v. that are Jacobian varieties of a curve with a non-trivial automorphism. A characterization in terms of hierarchies of p.d.e. for theta functions is also given.
Combinatorial geometry of the 2D Toda lattice and Davey Stewartson equation
研究二维托达晶格和戴维 - 斯图尔特森方程，借助Kodama和Williams受热带几何启发的算法方法，通过全非负格拉斯曼流形元素生成等高线图，推广其方法并在渐近情况完善前人工作。
We compute the group of automorphisms of an arbitrary ind-variety of (possibly isotropic) generalized flags. Such an ind-variety is a homogeneous ind-space for one of the ind-groups $SL(\infty)$, $O(\infty)$ or $Sp(\infty)$. We show that the respective automorphism groups are much larger than $SL(\infty)$, $O(\infty)$ or $Sp(\infty)$, and present the answer in terms of Mackey groups. The latter are groups of automorphisms of nondegenerate pairings of (in general infinite-dimensional) vector spaces. An explicit matrix form of the automorphism group of an arbitrary ind-variety of generalized fla…
In this paper we consider a reducible degeneration of a hyperelliptic curve of genus $g$. Using the Sato Grassmannian we show that the limits of hyperelliptic solutions of the KP-hierarchy exist and become soliton solutions of various types. We recover some results of Abenda who studied regular soliton solutions corresponding to a reducible rational curve obtained as a degeneration of a hyperelliptic curve. We study singular soliton solutions as well and clarify how the singularity structure of solutions is reflected in the matrices which determine soliton solutions.
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