On Fermat curves and maximal nodal curves

Abstract

Let f(x) = a(x − α1) · · · (x − αn), a > 0, be a real polynomial with n distinct real roots; it has [(n − 1)/2] maxima and (n − 1) − [(n − 1)/2] minima. Thom has studied the space of real polynomials and showed, for example, that any given polynomial f can be deformed into a special polynomial that has the same maxima and minima [8]. A typical such polynomial is the Chebyshev polynomial. A nodal curve C is an irreducible plane curve of degree n that contains only nodes (= A1 singularities). A nodal curve is called a maximal nodal curve if it is rational and nodal; by Plucker’s formula, it must contain (n−1)(n−2) 2 nodes to be maximal. In the space of polynomials of two variables, a maximal nodal curve can be understood as a generalization of a Chebyshev polynomial. In [6] the author constructed a maximal nodal curve of join type f(x)+ g(y) = 0 using a Chebyshev polynomial f(x) and a similar polynomial g(y) that has one maximal value and two minimal values. In this paper we present another extremely simple way, for a given integern > 2, to construct a beautifully symmetric and maximal nodal curve Dn as a by-product of the geometry of the Fermat curve xn+1 + y n+1 + 1 = 0. A smooth point P of a plane curve C is called a flex point of flex-order k ≥ 3 if the tangent line TP at P and C intersect with intersection multiplicity k. The maximal nodal curve Dn, which we construct in this paper, contains three flexes of flex-order n, and it is symmetric with respect to the permutation of three variables U,V,W. By a special case of Zariski [10] and Fulton [2], π1(P − C) = Z/nZ if C is a maximal nodal curve of degree n. The examples Dn provide an alternate proof. Zariski [10] observed that the fundamental group of the complement of an irreducible curve C of degree n is abelian if C has a flex of flex-order either n or n− 1. Since the moduli of maximal nodal curves of degree n is irreducible (by Harris [3]), the claim follows. For the construction, we start from the Fermat curve Fn : x + y n +1 = 0 and study singularities of the dual curve Fn. The Fermat curve and the dual curve Fn have canonical Z/nZ×Z/nZ actions, so the defining polynomial of Fn is written as h(u, v) = 0 for a polynomial h(u, v) of degree n− 1. The curve h(u, v) = 0 defines our maximal nodal curve Dn−1. Geometrically this is the quotient of the