Canonical Projective Embeddings of the Deligne–Lusztig Curves Associated to2A2,2B2, and2G2

Canonical Projective Embeddings of the Deligne-Lusztig Curves Associated to 2 A 2 , 2 B 2 and 2 G 2 Daniel M. Kane May 15, 2015 Abstract The Deligne-Lusztig varieties associated to the Coxeter classes of the algebraic groups 2 A 2 , 2 B 2 and 2 G 2 are affine algebraic curves. We produce explicit projective models of the closures of these curves. Furthermore for d the Coxeter number of these groups, we find polynomials for each of these models that cut out the F q -points, the F q d -points and the F q d+1 - points, and demonstrate a relation satisfied by these polynomials. Introduction There are four kinds of finite groups of Lie type of rank 1. The associated Deligne-Lusztig varieties for the Coxeter classes of these groups all give affine algebraic curves. The completions of these curves have several applications in- cluding the representation theory of the associated group ([1, 5]), coding theory ([4]) and the construction of potentially interesting covers of P 1 ([3]). In this paper, we consider these curves associated to the groups G = 2 A 2 , 2 B 2 and 2 G 2 . The remaining curve is associated to G = A 1 and is P 1 , but we do not cover this case as it is easy and doesn’t follow many of the patterns found in the analysis of the other three cases. For each of these curves, we explicitly construct an embedding C ,→ P(W ) where W is a representation of G of dimension 3,5, or 14 respectively, and provide an explicit system of equations cutting out C. The curve associated to 2 A 2 is the Fermat curve. The curve associated to B 2 is also well-known though not immediately isomorphic to our embedding. Embeddings of the curve associated to 2 G 2 were not known until recently. In [6], they constructed an explicit curve with the correct genus, symmetry group and number of points. Later, in [2], Eid and Duursma use this description to independently arrive at the embedding we produce in this paper. ∗ University of California, San Diego, Department of Mathematics / Department of Computer Science and Engineering, 9500 Gilman Drive #0404, La Jolla, CA 92093 dakane@ucsd.edu