Geometric nilpotent Lie algebras and zero-dimensional simple complete intersection singularities

The Levi theorem tells us that every finite-dimensional Lie algebra is the semi-direct product of a semi-simple Lie algebra and a solvable Lie algebra. Brieskorn gave the connection between simple Lie algebras and simple singularities. Simple Lie algebras have been well understood, but not the solvable (nilpotent) Lie algebras. Therefore, it is important to establish connections between singularities and solvable (nilpotent) Lie algebras. In this paper, we give a new connection between nilpotent Lie algebras and nilradicals of derivation Lie algebras of isolated complete intersection singularities. As an application, we obtain the correspondence between the nilpotent Lie algebras of dimension less than or equal to 7 and the nilradicals of derivation Lie algebras of isolated complete intersection singularities with modality less than or equal to 1. Moreover, we give a new characterization theorem for zero-dimensional simple complete intersection singularities.

Algebraic varieties with simple singularities related to some reflection groups

Related to a Coxeter group are certain sets of tangents of the deltoid with evenly distributed orientations forming simplicial line configurations. These configurations are used to construct curves and surfaces with [Formula: see text] singularities. Other surfaces associated with invariants of exceptional complex reflection groups are considered. A new lower bound for the maximal number of [Formula: see text] singularities in a sextic surface is obtained. Several Calabi–Yau threefolds defined as double coverings of the complex projective 3-space branched along nodal octic surfaces and Calabi–Yau quintic threefolds are analyzed. The Hodge numbers of a small resolution of all the nodes of the singular threefolds are obtained.

The 2-Component BKP Grassmannian and Simple Singularities of Type D

It was proved in 2010 that the principal Kac–Wakimoto hierarchy of type $D$ is a reduction of the 2-component BKP hierarchy. On the other hand, it is known that the total descendant potential of a singularity of type $D$ is a tau-function of the principal Kac–Wakimoto hierarchy. We find explicitly the point in the Grassmannian of the 2-component BKP hierarchy (in the sense of Shiota) that corresponds to the total descendant potential. We also prove that the space of tau-functions of Gaussian type is parametrized by the base of the miniversal unfolding of the simple singularity of type $D$.

Hypersurfaces with linear type singular loci

In this paper, necessary and sufficient criteria for the Jacobian ideal of a reduced hypersurface with isolated singularity to be of linear type are presented. We prove that the gradient ideal of a reduced projective plane curve with simple singularities ([Formula: see text]) is of linear type. We show that any reduced projective quartic curve is of gradient linear type.