On embedding of linear hypersurfaces

TL;DR

This paper investigates the embedding properties of linear hypersurfaces over fields, addressing longstanding questions in affine algebraic geometry using K-theory and group actions, and providing new results and counterexamples across different characteristics.

Contribution

It extends previous work on linear hypersurfaces by analyzing a broader class using advanced algebraic tools, and proves new cases of the Abhyankar Sathaye conjecture and counterexamples to the Zariski Cancellation Problem.

Findings

When characteristic is zero, certain hyperplanes are confirmed as coordinates.

The paper provides counterexamples to the Zariski Cancellation Problem in positive characteristic.

It establishes new families of hyperplanes satisfying the Embedding and Characterization problems.

Abstract

Linear hypersurfaces over a field $k$ have been playing a central role in the study of some of the challenging problems on affine spaces. Breakthroughs on such problems have occurred by examining two difficult questions on linear polynomials of the form $H:=\alpha(X_1,\dots,X_m)Y - F(X_1,\dots, X_m,Z,T)\in D:=k[X_1,\ldots,X_m, Y,Z,T]$: (i) Whether $H$ defines a closed embedding of $\mathbb{A}^{m+2}$ into $\mathbb{A}^{m+3}$, i.e., whether the affine variety $\mathbb{V}\subseteq \mathbb{A}^{m+3}_k$ defined by $H$ is isomorphic to $\mathbb{A}^{m+2}_k$. (ii) If $H$ defines a closed embedding $\mathbb{A}^{m+2}\hookrightarrow \mathbb{A}^{m+3}$ then whether $H$ is a coordinate in $D$. Question (i) connects to the Characterization Problem of identifying affine spaces among affine varieties; Question (ii) is a special case of the formidable Embedding Problem for affine spaces. In their earlier work the first two authors had addressed these questions when $\alpha$ is a monomial of the form $\alpha(X_1,\ldots,X_m) = X_1^{r_1}\dots X_m^{r_m}$; $r_i>1, 1 \leqslant i \leqslant m$ and $F$ is of a certain type. In this paper, using $K$-theory and $\mathbb{G}_a$-actions, we address these questions for a wider family of linear varieties. In particular, we obtain certain families of higher dimensional hyperplanes $H$ satisfying the Abhyankar Sathaye conjecture on the Embedding problem. For instance, we show that when the characteristic of $k$ is zero, $F \in k[Z,T]$ and $H$ defines a hyperplane, then $H$ is a coordinate in $D$ along with $X_1, X_2, \dots, X_m$. Our results in arbitrary characteristic yield counterexamples to the Zariski Cancellation Problem in positive characteristic.