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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 $R^G$?

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