On a ring of modular forms related to the Theta gradients map in genus 2

The level moduli space $A_g^{4,8}$ is mapped to the projective space by means of gradients of odd Theta functions, such a map turning out no to be injective in the genus 2 case. In this work a congruence subgroup $\Gamma$ is located between $\Gamma_2(4,8)$ and $\Gamma_2(2,4)$ in such a way the map factors on the related level moduli space $A_{\Gamma}$, the new map being injective on $A_{\Gamma}$. Satake's compactification $\text{Proj}A(\Gamma)$ and the desingularization $\text{Proj}S(\Gamma)$ are also due to be investigated, since the map does not extend to the boundary of the compactification; to aim at this, an algebraic description is provided, by proving a structure theorem both for the ring of modular forms $A(\Gamma)$ and the ideal of cusp forms $S(\Gamma)$