2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/165417Let k be an algebraically closed field, let K/k be a finitely generated field extension of transcendence degree 2 with automorphism sigma, and let A be an N-graded subalgebra of Q = K[t; sigma] with A_n finite dimensional over k for all n. Then if A is big enough in Q in an appropriate sense, we prove that GKdim A = 3,4,5 or is infinite, with the exact value depending only on the geometric properties of sigma. The proof uses techniques in the birational geometry of surfaces which are of independent interest.26 pages, minor corrections from previous version based on referee report. To appear in Trans. Amer. Math. SocRings and Algebras14A22, 14E05, 16P90, 16S38, 16W50text