On stabilizers of algebraic function fields of one variable

Let $\tilde K$ be a fixed algebraic closure of an infinite field K. We consider an absolutely integral curve Γ in $\mathbb{P}_{K}^{n}$ with n ≥ 2. The curve $\it\Gamma_{\tilde{K}}$ should have only finitely many inflection points, finitely many double tangents, and there exists no point in $\mathbb{P}_{\tilde{K}}^{n}$ through which infinitely many tangents to $\it\Gamma_{\tilde{K}}$ go. In addition there exists a prime number q such that $\it\Gamma_{\tilde{K}}$ has a cusp of multiplicity q and the multiplicities of all other points of $\it\Gamma_{\tilde{K}}$ are at most q. Under these assumptions, we construct a non-empty Zariski-open subset O of $\mathbb{P}_{\tilde{K}}^{n}$ such that if n ≥ 3, the projection from each point o ∈ O(K) birationally maps Γ onto an absolutely integral curve Γ′ in $\mathbb{P}_{K}^{n-1}$ with the same properties as Γ (keeping q unchanged). If n = 2, then the projection from each o ∈ O(K) maps Γ onto $\mathbb{P}_{K}^{1}$ and leads to a stabilizing element t of the function field F of Γ over K. The latter means that F/K(t) is a finite separable extension whose Galois closure ${\hat F}$ is regular over K.