Threefolds from Solutions of a Partial Differential Equation

ABSTRACT

Several types of complex algebraic threefolds are constructed. They are based on solutions of a second-order linear partial differential equation. The solutions are related to a one-parameter family of polynomials associated with configurations of real lines in the plane. Some of the polynomials produce hypersurfaces with many singularities. Particular cases of quintic and sextic threefolds are analyzed. The construction of Calabi–Yau threefolds is based on special types of the line configurations.

KEYWORDS:

• algebraic varieties
• singularities
• multivariate polynomials

2000 AMS SUBJECT CLASSIFICATION:

• Primary 14J30
• Secondary 14J17
• 35L10

Acknowledgment

The author thanks the anonymous referees whose comments helped to improve the presentation of the results.