Standard Norm Varieties for Milnor Symbols mod p

TL;DR

This paper establishes birational equivalences between standard norm varieties for Milnor symbols mod p and classical algebraic varieties, clarifying their geometric structure for different primes and lengths.

Contribution

It proves birational isomorphisms of standard norm varieties with classical varieties for specific cases and conjectures a general equivalence.

Findings

Standard norm varieties for p=2 are birationally isomorphic to Pfister quadrics.

For p>2, n=2, they are birationally isomorphic to Severi-Brauer varieties.

The varieties for p>2, n=3 are related to reduced norms of cyclic algebras.

Abstract

We prove that the standard norm varieties for Milnor symbols mod p of length n are birationally isomorphic to Pfister quadrics when p = 2, to Severi-Brauer varieties when p > 2, n = 2, and to varieties defined by reduced norms of cyclic algebras when p > 2, n = 3. In the case p = 2 and the case p > 2, n = 2, the results imply that the standard norm varieties for two equal Milnor symbols mod p are birationally isomorphic, and we conjecture this in general.