[Paper Review] Birkhoff strata of Sato Grassmannian and algebraic curves

B. G. Konopelchenko, G. Ortenzi|arXiv (Cornell University)|May 12, 2010

Nonlinear Waves and Solitons56 references3 citations

TL;DR

This paper establishes a geometric and algebraic correspondence between Birkhoff strata of the Sato Grassmannian and families of algebraic curves by identifying subbundles of the tautological bundle that are closed under pointwise multiplication. It shows that each stratum Σ_S contains a subset W_S where the fiber forms an infinite-dimensional commutative algebra, geometrically corresponding to families of algebraic curves—such as Veronese curves in the big cell, elliptic curves in Σ₁, and (n+1,n+2) plane curves in higher strata—offering a new intrinsic link between integrable systems and algebraic geometry.

ABSTRACT

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) curves of all degrees. For $W_1$ such tautological subbundle is the family of coordinate rings for elliptic curves. For higher strata, the subbundles $TB_{W_{1,2,\dots,n}}$ represent families of plane $(n+1,n+2)$ curves (trigonal curves at $n=2$) and space curves of genus $n$. Two methods of regularization of singular curves contained in $TB_{W_{\hat{S}}}$, namely, the standard blowing-up and transition to higher strata with the change of genus are discussed.

Motivation & Objective

• To explore the intrinsic geometric and algebraic structures of algebraic curves within the Sato Grassmannian itself, rather than as external data.
• To identify subsets W_S within each Birkhoff stratum Σ_S such that the fibers of the tautological subbundle TB_{W_S} are closed under pointwise multiplication.
• To demonstrate that these fibers form infinite families of infinite-dimensional commutative associative algebras, geometrically corresponding to families of algebraic curves.
• To investigate the role of singular curves in these families and propose regularization methods such as blowing-up and stratum transitions.

Proposed method

• Analyzes the tautological subbundle TB_{W_S} over subsets W_S in Birkhoff strata Σ_S of the Sato Grassmannian.
• Identifies conditions under which fibers of TB_{W_S} are closed under pointwise multiplication, making them commutative associative algebras.
• Uses elementary algebraic geometry and Sato’s framework to connect the structure of these algebras to known families of algebraic curves.
• Applies the Krichever map and its inverse to relate subspaces in the Grassmannian to algebraic curves.
• Introduces two regularization techniques: standard blow-up within the same stratum and transition to higher strata with changed genus.
• Employs index theory via the invariant (∂̄_{W_S}) to classify the strata, with values −n for Σ_{1,2,…,n}.

Experimental results

Research questions

• RQ1 How can algebraic curves be naturally embedded within the Sato Grassmannian via its tautological bundle?
• RQ2 What algebraic and geometric properties do fibers of the tautological subbundle TB_{W_S} possess when closed under pointwise multiplication?
• RQ3 What types of algebraic curves arise in different Birkhoff strata, particularly in the big cell and strata Σ₁, Σ_{1,2}, and higher Σ_{1,2,…,n}?
• RQ4 How can singular curves within these families be regularized, and what are the implications of such regularization?
• RQ5 What is the role of the index (∂̄_{W_S}) in classifying the geometric and algebraic structure of these curve families?

Key findings

• The big cell Σ_∅ corresponds to the tautological subbundle TB_{W_∅}, which forms a tower of families of normal rational (Veronese) curves of all degrees ≥2.
• For the stratum Σ₁, the tautological subbundle TB_{W₁} corresponds to the coordinate rings of elliptic curves, with index (∂̄_{W₁}) = −1.
• In the stratum Σ_{1,2}, TB_{W_{1,2}} contains families of plane trigonal curves of genus 2, and the index is (∂̄_{W_{1,2}}) = −2.
• For higher strata Σ_{1,2,…,n}, the subbundle TB_{W_{1,2,…,n}} contains families of plane (n+1,n+2) curves of genus n, with index (∂̄_{W_{1,2,…,n}}) = −n.
• Projections of basic curves to lower-dimensional subspaces yield singular higher-degree curves, which can be regularized via blow-up or transition to higher strata.
• The paper conjectures a systematic correspondence between Birkhoff strata and families of algebraic curves with increasing genus and negative index −n.