A mixed curve is a subset of \(\mathbb{C}^2\) defined by the vanishing of a polynomial \(f(z,\overline{z})=0\), where \(\overline{z}\) is the complex conjugate of \(z \in \mathbb{C}^2\). This interesting paper develops the basic intersection theory of mixed curves, generalizing the main results of the holomorphic case.
Let \(P\) be an isolated intersection point of mixed curves \(C\) and \(C'\), and suppose that both mixed curves have at worst an isolated mixed singularity at \(P\). The author defines the local intersection number \(I_{\text{top}}(C,C';P)\) from the defining equations of \(C\) and \(C'\), and shows that the local intersection number is stable under a bifurcation of \(P\). It follows that, first, the definition of the local intersection number coincides with the usual one for complex analytic curves, and second, if \(C\) and \(C'\) are mixed projective curves, intersecting at points \(P_1,\dots,P_\mu\) at which both mixed curves have at worst an isolated mixed singularity, then \(\sum_{j=1}^\mu I_{\text{top}}(C,C';P_j)=[C] \cdot [C']\). The existence and the computation of the fundamental class \([C]\) was treated in the author’s previous paper [“On mixed projective curves”, arXiv:0910.2523].
The last section is devoted to a detailed study of the roots of a mixed polynomial \(h(u,\overline{u})\) of one variable \(u\) as a special case.
This paper is clearly written and contains a number of good examples.