Projective normality of model varieties and related results

TL;DR

This paper proves the projective normality of model varieties and related compactifications, establishing surjectivity of section multiplication and confirming conjectures for specific orbits, advancing understanding in algebraic geometry.

Contribution

It introduces a general method to prove surjectivity of section multiplication on model varieties, leading to normality results and confirming conjectures in specific cases.

Findings

Multiplication of sections is always surjective on model wonderful varieties.

The cones over these varieties are normal.

Confirmed conjectures for E8-type model orbits.

Abstract

We prove that the multiplication of sections of globally generated line bundles on a model wonderful variety M of simply connected type is always surjective. This follows by a general argument which works for every wonderful variety and reduces the study of the surjectivity for every couple of globally generated line bundles to a finite number of cases. As a consequence, the cone defined by a complete linear system over M or over a closed G-stable subvariety of M is normal. We apply these results to the study of the normality of the compactifications of model varieties in simple projective spaces and of the closures of the spherical nilpotent orbits. Then we focus on a particular case proving two specific conjectures of Adams, Huang and Vogan on an analogue of the model orbit of the group of type E8.