On a question of Dolgachev (Q2183381)

The authors define a rational self-map \(H_n\) on the space of plane curves of even degree \(n\), over a field \(k\) of characteristic coprime to 6, using classical contravariants. They show that for \(n=4\) the map \(H_4:\mathbb{P}_k^{14}\dashrightarrow \mathbb{P}_k^{14}\) is generically finite and thus descends to a generically finite rational map on the GIT-quotient of \(\mathbb{P}_k^{14}\) by \(\mathbb{P}GL_3\), \(\overline{H}_4:\mathbb{P}_k^{14}/\!/\mathbb{P}GL_3\dashrightarrow \mathbb{P}_k^{14}/\!/\mathbb{P}GL_3,\) having the same degree of \(H_4,\) which they prove to be 15, as previously computed by the second author. Let us recall that \(\mathbb{P}^{14}/\!/\mathbb{P}GL_3\) is birational to the moduli space \(\mathcal{M}_3\) of smooth curves of genus 3. This paper was indeed inspired by \textit{I. V. Dolgachev} who studied rational self-maps of moduli spaces of curves of low genus and of hypersurfaces in [Pure Appl. Math. Q. 12, No. 3, 335--352 (2016; Zbl 1387.14050)].