Explicit constructions for genus 3 Jacobians
Let \(X\) be a nonsingular curve of genus three defined by a quartic \(F\) in \(\mathbb{P}^2\). Assume that \(X\) has a point \(p\) such that its tangent line intersects \(X\) at \(p\) with multiplicity four. Under this assumption the authors show in this paper that there exists a bijection between an open subset of the jacobian variety of \(X\) and the variety \(\{(A, B, G, H)\in H^0(\mathcal{O}_{\mathbb{P}^2}(2))^4;F=AG-BH\}\) with \(A\) and \(B\) of the form \(A=a_{00}+a_{10}x+a_{01}y-x^2, B=b_{00}+b_{10}x+b_{01}y-xy\), and \(H\) without the term \(x^2\). Furthermore they provide us with an explicit form of the morphism associated with the multiplication by -1, and a set of equations for the Kummer variety of the jacobian variety.
scientific article; zbMATH DE number 6368770