Geometry of plane algebraic curves

设C是复数域上的d次不可约平面曲线。对于C上的一个非常数有理函数Φ，我们可以把一个从C的非奇异模型到P^1的态射Φ联系起来。用G(或Gon(C))表示的C的次性被定义为这种态射的最小次数。设ν表示C的奇点的最大重数，在这种情况下，我们说C是(d，ν)型的。然后我们很容易看到G≦d-ν。首席调查员和他的学生M.Ohkouhi证明了两种相等的标准：G=d-ν(东京数学杂志，2004年)。然而，对于许多平面曲线来说，等价性并非如此。首席调查员还证明了G&lt；d-ν案件的G的两个下界，并讨论了各种例子(预印本、提交中)。最近，他得到了C的亏格g与性度G之间的关系。更准确地说，如果g≦B(d，ν)，则等式G=d-ν成立。首席研究员和他的学生M.Saleem将有理平面曲线C分类为(d，d-2)型(Saitama Math.J.27,2005)。特别地，他们提供了构造这种曲线的归纳算法，并证明了任何这样的曲线C都可以通过Cremona变换变换成一条直线。以前，只有尖点的(d，d-2)型有理平面曲线被分类。为了描述多分支平面曲线奇点，引入了多重序列系的概念。将这些结果推广到任意亏格的(d，d-2)型平面曲线，即椭圆曲线和超椭圆曲线(预印本)。

Let C be an irreducible plane curve of degree d over the complex number field. To a non-constant rational function Φ on C, we can associate a morphism Φfrom the non-singular model of C to P^1. The gonality of C, denoted by G (or Gon(C)), is defined to be the minimum of the degrees of such morphisms. Let ν denote the maximal multiplicity of the singular points of C. In this situation, we say that C is of type (d,ν). We then easily see that G≦d-ν. The head investigator and his student M. Ohkouchi proved two kinds of criteria for the equality : G=d-ν (Tokyo J.Math.J.2004). However, for many plane curves, the equality is not the case. The head investigator also proved two lower bounds for G for the case in which G<d-ν and discussed various kinds of examples (Preprint, under submission). More recently, he obtained a relation between the genus g of C and the gonality G. More precisely, if g≦B (d,ν), then the equality G=d-ν holds.The head investigator and his student M. Saleem classified rational plane curves C of type (d, d-2) (Saitama Math.J.27,2005). In particular, they provide an inductive algorithm to construct such curves and proved that any such curve C is transformable into a line by a Cremona transformation. Previously, rational plane curves of type (d, d-2) with only cusps were classified. To describe multi-branched plane curve singularities, the notion of the system of multiplicity sequences were introduced. These results are generalized to type (d, d-2) plane curves with arbitrary genus, which are elliptic and hyperelliptic curves (Preprint).