2025/04/30 by Gene Freudenburg, Freudenburg, Gene · 1 citation
Mathematics · #14R20 #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Algebraic structures and combinatorial models #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology
paper · pdf · doi:10.48550/arxiv.2504.21712
openalex publication_date 2025/04/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
For an algebraically closed field k of characteristic zero and a linear algebraic k-group G, it is well known that every affine G-variety admits a G-equivariant closed embedding into a finite-dimensional G-module. Such an embedding is a presentation of the G-variety, and a minimal presentation is one for which the dimension the G-module is minimal. The problem of finding a minimal presentation generalizes the problem of determining whether a group action on affine space is linearizable. We give a minimal presentation for each homogeneous space for SL2(k). This constitutes the paper's main work. Of particular interest are the surfaces Y=SL2(k)/T and X=SL2(k)/N where T is the one-dimensional torus and N is its normalizer. We show that the minimal presentation of X has dimension 5, the embedding dimension of X is 4, and there does not exist a closed SL2-equivariant embedding of X in Ak4. Thus, the SL2-action on X is absolutely nonextendable to Ak4. We give two other examples of surfaces with absolutely nonextendable group actions. In addition, X is noncancelative, that is, there exists a surface Z such that X× Ak1≅k Z× Ak1 and X\not≅kZ. Finally, we settle the long-standing open question of whether there exist inequivalent closed embeddings of Y in Ak3 by constructing inequivalent embeddings.