Singular points of complex algebraic hypersurfaces

Irina A. Antipova ; Evgeny N. Mikhalkin ; Avgust K. Tsikh

Žurnal Sibirskogo federalʹnogo universiteta. Matematika i fizika, Tome 11 (2018) no. 6, pp. 670-679

Résumé

We consider a complex hypersurface $V$ given by an algebraic equation in $k$ unknowns, where the set $ A\subset {\mathbb Z}^k $ of monomial exponents is fixed, and all the coefficients are variable. In other words, we consider a family of hypersurfaces in $ ({\mathbb C \setminus 0}) ^ {k} $ parametrized by its coefficients $a =(a_{\alpha})_{\alpha \in A} \in {\mathbb C} ^{A} $. We prove that when $A$ generates the lattice $\mathbb Z^k$ as a group, then over the set of regular points $a$ in the $A$-discriminantal set, the singular points of $V$ admit a rational expression in $a$.

Détail

Keywords: singular point
Mots-clés : $A$-discriminant, logarithmic Gauss map.