Birkhoff strata of Sato Grassmannian and algebraic curves

Algebraic and geometric structures associated with Birkhoff strata of Sato Grassmannian are analyzed. It is shown that each Birkhoff stratum $\Sigma_S$ 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.Comment: 31 pages, no figures, version accepted in Journal of Nonlinear Mathematical Physics. The sections on the integrable systems present in previous versions has been published separatel

Permutohedral complexes and rational curves with cyclic action

We define a moduli space of rational curves with finite-order automorphism and weighted orbits, and we prove that the combinatorics of its boundary strata are encoded by a particular polytopal complex that also captures the algebraic structure of a complex reflection group acting on the moduli space. This generalizes the situation for Losev-Manin's moduli space of curves (whose boundary strata are encoded by the permutohedron and related to the symmetric group) as well as the situation for Batyrev-Blume's moduli space of curves with involution, and it extends that work beyond the toric context.Comment: 47 pages, 12 figure

Quatroids and Rational Plane Cubics

It is a classical result that there are $12$ (irreducible) rational cubic curves through $8$ generic points in $\mathbb{P}_{\mathbb{C}}^2$, but little is known about the non-generic cases. The space of $8$-point configurations is partitioned into strata depending on combinatorial objects we call quatroids, a higher-order version of representable matroids. We compute all $779777$ quatroids on eight distinct points in the plane, which produces a full description of the stratification. For each stratum, we generate several invariants, including the number of rational cubics through a generic configuration. As a byproduct of our investigation, we obtain a collection of results regarding the base loci of pencils of cubics and positive certificates for non-rationality.Comment: 34 pages, 11 figures, 5 tables. Comments are welcome