Projective plane curves and the automorphism groups of their complements
Let \(C\) be an irreducible algebraic curve of degree \(d\) on \({\mathbb{P}}^ 2={\mathbb{P}}^ 2({\mathbb{C}})\) and let \({\mathcal G}\) be the automorphism group of the open algebraic surface \({\mathbb{P}}^ 2-C\). In case \(d=1\), it is known that \({\mathcal G}\) is generated by linear transformations and de Jonquières transformations of \({\mathbb{P}}^ 2-C\). In case \(d=2\), the generators of a similar kind have been found by \textit{V. I. Danilov} and \textit{M. Kh. Gizatullin} [Math. USSR, Izv. 11, 51-98 (1977; translation from Izv. Akad. Nauk SSSR, Ser. Mat. 41, 54-103 (1977; Zbl 0357.14003)]. Here we study the structure of \({\mathcal G}\) and at the same time the property of C in the case when \(d\geq 3\). Let \({\mathcal L}\) be the linear part of \({\mathcal G}\), i.e., \(\{T\in Aut({\mathbb{P}}^ 2)| T(C)=C\}\) and let \(\Delta\) [resp. \(\Delta_ e]\) be the curve defined by \(XYZ=X^ 3+Y^ 3\) [resp. \(Y^ eZ^{d-e}=X^ d\), where \((e,d)=1\) and 1\(\leq e\leq d-2]\). Let N be the number of elements of \(f^{-1}(S)\), where \(f:\tilde C\to C\) is the normalization of C and S is the set of singular points of C. In case \(N=1\), we put \(R=d^ 2-(\sum^{p}_{i=1}e_ i\!^ 2)-e_ p+1,\) where \(e_ 1,...,e_ p\) are the multiplicities of all infinitely near singular points. Then the result is as follows: The order of \({\mathcal G}\) is finite if and only if C satisfies any one of the following three conditions. (1) \(C\) is not rational. (2) \(N\geq 2\) and \(C\) is projectively equivalent to neither \(\Delta\) nor \(\Delta_ e\). (3) \(N=1\) and \(R\leq - 1\). On the contrary, for the remaining curves with \(d\geq 3\), \({\mathcal G}\) has the following properties. (4) If \(C\) is projectively equivalent to \(\Delta_ e\), where \(e\geq 2\), then \({\mathcal G}={\mathcal L}\cong G_ m,\) which is the multiplicative group. (5) If \(C\) is projectively equivalent to \(\Delta\), then the order of \({\mathcal G}\) is countably infinite and \({\mathcal L}\) is the dihedral group of order 6. (6) If \(C\) is a rational curve with \(N=1\) and \(R\geq 0\), then \({\mathcal G}\supset(G_ a)^ n\) for every positive integer n, where \(G_ a\) is the additive group. In this class of curves the order of \({\mathcal L}\) is infinite if and only if \(C\) is projectively equivalent to \(\Delta_ 1.\) For the curve of (6) the structure of \({\mathcal G}\) seems to be complicated, but this curve has the next property. There are one or two irreducible curves \(C'\) and \(C''\) and two or three lines \(L_ i\), where \(i=1,2,3\), which satisfy the following: (1) In case \(R\neq 1\), there is an isomorphism \({\mathbb{P}}^ 2-(C\cup C')\cong {\mathbb{P}}^ 2-(L_ 1\cup L_ 2).\) (2) In case \(R=1\), there is an isomorphism \({\mathbb{P}}^ 2-(C\cup C'\cup C'')\cong {\mathbb{P}}^ 2-(L_ 1\cup L_ 2\cup L_ 3),\) where \(L_ 1\cap L_ 2=L_ 1\cap L_ 3.\) By this result we see that the logarithmic Kodaira dimension \({\bar \kappa}({\mathbb{P}}^ 2-C)=-\infty\) if \(C\) is a rational curve with \(N=1\) and \(R\geq 0\).
automorphism group of complement of plane algebraic curve
scientific article; zbMATH DE number 3849402