On piecewise isomorphism of some varieties

TL;DR

This paper proves that certain symmetric powers of complex affine and projective spaces are piecewise isomorphic to well-known varieties, revealing new structural insights into their stratifications.

Contribution

It establishes piecewise isomorphisms between symmetric powers of complex spaces and classical varieties, extending understanding of their geometric and combinatorial structures.

Findings

$S^m(C^n)$ is piecewise isomorphic to $C^{mn}$

$S^m(CP^ )$ is piecewise isomorphic to $Gr(m, )$

Provides new connections between symmetric powers and classical varieties

Abstract

Two quasi-projective varieties are called piecewise isomorphic if they can be stratified into pairwise isomorphic strata. We show that the m-th symmetric power $S^m(C^n)$ of the complex affine space $C^n$ is piecewise isomorphic to $C^{mn}$ and the m-th symmetric power $S^m(CP^\infty)$ of the infinite dimensional complex projective space is piecewise isomorphic to the infinite dimensional Grassmannian $Gr(m,\infty)$.