Geometry of the group of Weierstrass points of a smooth quartic (Q1609539)

Let \(C_t\) be the projective smooth curve which is birational to the curve \(y^4= x(x-1)(x-t)\), where \(t\not\in \{0,1\}\). We denote by \(W(C_t)\) the subgroup of the Jacobian of \(C_t\) generated by the Weierstrass points of \(C_t\). In this paper, the author proves that for every number field \(K\), there is a finite set \(S_K\) such that if \(t\in K\setminus S_K\), then \(W(C_t)\cong \mathbb{Z}^9\times (\mathbb{Z}/4\mathbb{Z})^2\). Moreover, he obtains bounds for the rank and the finite part of the case of a generic quartic depending on the number of hyperflexes.