Geometry of plane algebraic curves

设C是复数域上d次不可约平面曲线.对于C上的一个非常数有理函数Φ，我们可以将C的非奇异模型中的一个态射Φ与P^1联系起来。C的角性，记为G（或Gon（C）），定义为此类态射的度的最小值。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，v）型的。我们可以很容易地看到，G d-ν。首席调查员和他的学生M。Ohkouchi证明了两种标准的平等：G=d-ν（东京J.Math.J.2004）。然而，对于许多平面曲线，情况并非如此。首席研究员还证明了G<d-ν的情况下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与角度G之间的关系。更确切地说，如果g ≠ B（d，ν），则等式G=d-ν成立。Saleem将有理平面曲线C分类为（d，d-2）型（琦玉Math.J.27，2005）。特别是，他们提供了一个归纳算法来构造这样的曲线，并证明了任何这样的曲线C是可变换成一条线的克雷莫纳变换。以前，有理平面曲线的类型（d，d-2），只有尖点进行分类。为了描述多分支平面曲线的奇异性，引入了重数序列系的概念。这些结果被推广到任意亏格的（d，d-2）型平面曲线，即椭圆曲线和超椭圆曲线（Preprint）。

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).