We prove that the polar degree of an arbitrarily singular projective hypersurface can be decomposed as a sum non-negative numbers which represent local vanishing cycles two different types. This yields lower bounds for any hypersurface.

We prove that for any germ of complex analytic set in Cn there exists a hypersurface singularity whose Milnor fibration has trivial geometric monodromy and fibre homotopic to the complement of the germ of complex analytic set. As an application we show an example of a quasi-homogeneous hypersurface singularity, with trivial geometric monodromy and simply connected and non-formal Milnor fibre.

Using the Blaschke-Berwald metric and affine shape operator of a hypersurface M in (n+1)-dimensional real space we can define some generalized curvature tensor named Opozda-Verstraelen tensor. In this paper determine conditions pseudosymmetry type expressed by for locally strongly convex hypersurfaces M, n>2, with two distinct principal curvatures or three assuming that at least one has multipl...

This paper shows that the general hypersurface of degree ≥ 6 in projective four space cannot support an indecomposable rank two vector bundle which is Arithmetically CohenMacaulay and four generated. Equivalently, the equation of the hypersurface is not the Pfaffian of a four by four minimal skewsymmetric matrix.

In this paper we prove the birational superrigidity and nonrationality of a hypersurface X ⊂ P of degree 6 such that the hypersurface X does not contain three-dimensional linear subspaces of P and the only singularities of X are isolated ordinary double points.

We prove the non-rationality of a double cover of P branched over a hypersurface F ⊂ P of degree 2n having isolated singularities such that n ≥ 4 and every singular points of the hypersurface F is ordinary, i.e. the projectivization of its tangent cone is smooth, whose multiplicity does not exceed 2(n− 2).

Constructions of metrics with special holonomy by methods of exterior differential systems are reviewed and the interpretations of these construction as ‘flows’ on hypersurface geometries are considered. It is shown that these hypersurface ‘flows’ are not generally well-posed for smooth initial data and counterexamples to existence are constructed.

This paper studies geometric properties of the Iterated Matrix Multiplication polynomial and the hypersurface that it defines. We focus on geometric aspects that may be relevant for complexity theory such as the symmetry group of the polynomial, the dual variety and the Jacobian loci of the hypersurface, that are computed with the aid of representation theory of quivers.

We give an upper bound for the minimal discrepancies of hypersurface singularities. As an application, we show that Shokurov’s conjecture is true for log-terminal threefolds.

Let (9,+1 be the ring of germs of holomorphic functions (C ~+ 1, 0 ) ~ C. There are many important equivalence relations that have been defined on the elements of (9+ 1. ~ ' , ~s and ~f-equivalence are well known in function theory. Each of these equivalence relations can be defined in terms of a Lie group action on (9 +1For instance two functions are defined to be ~-equivalent if they are the ...