Minimization of the embeddings of the curves into the affine plane (Q2565387)

Let \(C\) be a smooth affine algebraic curve with only one place at infinity defined over an algebraically closed field \(k\) of characteristic zero; we also call \(C\) a once punctured smooth algebraic curve. Assume that \(C\) is embedded into the affine plane \(\mathbb{A}^2\) as a closed curve. The image of \(C\) by an algebraic automorphism of \(\mathbb{A}^2\) is again a curve of the same nature as \(C\). One may then ask what is the smallest among the degrees of \(\varphi(C)\) when \(\varphi\) ranges over automorphisms of \(\mathbb{A}^2\). We say that \(\varphi (C)\) is a minimal embedding of \(C\) if the degree of \(\varphi (C)\) is the smallest. -- Our theorem is the following: Let \(C\) be a once punctured smooth algebraic curve of genus \(g\), which is embedded into the affine plane \(\mathbb{A}^2 =\text{Spec} k [x,y]\) as a closed curve defined by \(f(x,y)=0\). Then there exist new coordinates \(u,v\) of \(\mathbb{A}^2\) such that (1) \(k[x,y] =k[u,v]\), and (2) \(h(u,v): =f(x(u,v),\;y(u,v))\) and \(e:=\deg h(u,v)\) are given as follows if \(g\leq 4\) case \(g=0\): \(e=1\) and \(h=u\). case \(g=1\): \(e=3\) and \(h=v^2- (u^3+ au+b)\) with \(a,b\in k\). case \(g=2\): \(e=5\) and \(h=v^2- (u^5+ au^3+ bu^2+ cu+d)\) with \(a,b,c\), \(d\in k\). case \(g=3\): there are three types (a) \(e=4\) and \(h=v^3+g_1(u)v-(u^4+g_2(u))\) with \(g_i(u)\in k[u]\) and \(\deg g_i(u)\leq 2\) for \(i=1,2\). (b) \(e=7\) and \(h=v^2-(u^7+g(u))\) with \(g(u)\in k[u]\) and \(\deg g(u)\leq 5\). (c) \(e=6\) and the multiplicity sequence of singularities at the point at infinity \(P_0\) is \((2^7)\), where \((2^7)\) implies that there are 7 double points centered at \(P_0\). case \(g=4\): there are four types: (a) \(e=5\) and \(h=v^3+g_1(u)v-(u^5+g_2(u))\) with \(g_i(u)\in k[u]\) and \(\deg g_i(u)\leq 3\) for \(i=1,2\). (b) \(e=9\) and \(h=v^2-(u^9+g(u))\) with \(g(u)\in k[u]\) and \(\deg g(u)\leq 7\). (c) \(e=6\) and the multiplicity sequence of singularities at the point at infinity is \((2^6)\). (d) \(e=9\) and the multiplicity sequence of singularities is \((3^8)\). (3) The automorphism \(\varphi\) of \(\mathbb{A}^2\) induced by \(^a\varphi(x)=x(u,v)\) and \(^a\varphi (y) =y(u,v)\) is described explicitly as a Cremona transformation of \(\mathbb{P}^2\) induced by \(\varphi\).