Projective Singular Hypersurfaces and Milnor Algebras

The aim of this book is to investigate some objects from Algebraic Geometry, namely the singular projective hypersurfaces, using primarily the technique of graded Milnor algebras and some related invariants. It is interesting to explore the potential relations between the coefficients of the Hilbert-Poincare series associated to a singular hypersurface and some properties of the hypersurface (like genus, type of singularities, Milnor and Tjurina numbers, rigidity, free divisors and syzygies of associated Jacobian ideal). For these Hilbert-Poincare series, we introduce new invariants to understand and quantify the difference between such a series and the series associated to a smooth hypersurface, say of Fermat type. Another main idea consists in defining nodal curves based on orthogonal polynomials: Chebyshev, Hermite, Laguerre and Legendre. We present an extension of Milnor algebra, in two dimensions, for two hypersurfaces. To compute easily and make large scale simulations, we list our procedures, written in the Singular language. We present in this book many conjectures and we invite the reader to prove them or try to generalize them in higher dimension.