Canonical dimension of projective PGL_1(A)-homogeneous varieties

TL;DR

This paper investigates the canonical dimension of projective PGL_1(A)-homogeneous varieties, reducing the problem to generalized Severi-Brauer varieties, and establishes that for certain cases, the canonical dimension equals the variety's dimension.

Contribution

It provides a reduction method for computing canonical p-dimension and proves equality of canonical dimension and dimension for specific cases involving powers of 2.

Findings

Canonical 2-dimension equals the variety's dimension for ind A = 2e power of 2.

Reduction to generalized Severi-Brauer varieties simplifies the computation.

Results apply to projective PGL_1(A)-homogeneous varieties with ind A as a power of p.

Abstract

Let A be a central division algebra over a field F with ind A = n. In computing canonical p-dimension of projective PGL_1(A)-homogeneous varieties, for p prime, we can reduce to the case of generalized Severi-Brauer varieties X_e(A) with ind A a power of p divisible by e. We prove that canonical 2-dimension (and hence canonical dimension) equals dimension for all X_e(A) with ind A = 2e a power of 2.