Geometry of plane algebraic curves

设C为复数域上的一条不可约的d次平面曲线。对于C上的非常有理函数Φ，我们可以将C的非奇异模型的态射Φfrom与P^1联系起来。C的向性，用G（或Gon(C)）表示，被定义为这些态射的最小度。设ν表示C的奇异点的最大多重数。在这种情况下，我们说C的类型是（d,ν）。我们很容易看到G≦d-ν。首席研究员和他的学生M. Ohkouchi证明了两种等式准则：G=d-ν (Tokyo J.Math.J.2004)。然而，对于许多平面曲线来说，等式不是这样的。首席研究员还证明了G<d-ν的两个下界，并讨论了各种例子（预印本，提交）。最近，他得到了C的g属与g的gonality之间的关系。更准确地说，如果g≦B (d,ν)，则等式g =d-ν成立。首席研究员及其学生M. Saleem对（d, d-2）型的有理平面曲线C进行了分类（埼玉数学，j .27,2005）。特别地，他们提供了一种构造此类曲线的归纳算法，并证明了任意此类曲线C都可以通过Cremona变换变换成直线。以前，只对（d, d-2）型的有理平面曲线进行尖点分类。为了描述平面曲线的多分支奇异性，引入了多重序列系统的概念。这些结果推广到具有任意属的（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).