$\mathbb{K}$-uniruled sets in affine geometry

TL;DR

This thesis investigates $\\mathbb{K}$-uniruled sets in affine geometry, establishing bounds on their degree and exploring their occurrence in fixed point sets of algebraic group actions.

Contribution

It introduces bounds on the degree of $\\mathbb{K}$-uniruledness for non-properness sets of polynomial maps and relates $\\mathbb{K}$-uniruledness to fixed points of algebraic groups.

Findings

Bound on the degree of $\\mathbb{K}$-uniruledness for non-properness sets

Equivalence conditions for $\\mathbb{K}$-uniruledness

Identification of $\\mathbb{K}$-uniruled sets among fixed points

Abstract

The main goal of this thesis is to study $\mathbb{K}$-uniruled sets that appear in affine geometry. At the beginning we discuss the property of $\mathbb{K}$-uniruledness and its equivalent conditions. Then we bound from above the degree of $\mathbb{K}$-uniruledness of the non-properness set of a polynomial map in terms of its degree and degree of $\mathbb{K}$-uniruledness of the domain variety. At the end we show that some sets associated with the set of fixed points of an algebraic group action are $\mathbb{K}$-uniruled.