# Maximum degree of the generators of the ring of invariant sections a line bundle

Let $$X$$ be a smooth projective geometrically connected curve over $$\mathbb{F}_2$$ endowed with a theta characteristic $$D$$. Consider the graded $$\mathbb{F}_2$$-algebra $$R$$ of sections of tensor powers of $$\mathcal{O}(D)$$. I haven't quite figured out what is the optimal bound but I think there is an absolute bound on the degree of generators and relations in $$R$$. If you read the paper https://arxiv.org/abs/1507.02643 carefully you might be able to extract the optimal bound.

Now suppose the group $$GL_2(\mathbb{F}_3)$$ (order 12) is acting by automorphisms on $$X$$. Is there an absolute bound on the degree of generators and relations in $$R^G$$?