Towards the classification of symplectic linear quotient singularities admitting a symplectic resolution

Over the past 2 decades, there has been much progress on the classification of symplectic linear quotient singularities V/G admitting a symplectic (equivalently, crepant) resolution of singularities. The classification is almost complete but there is an infinite series of groups in dimension 4—the symplectically primitive but complex imprimitive groups—and 10 exceptional groups up to dimension 10, for which it is still open. In this paper, we treat the remaining infinite series and prove that for all but possibly 39 cases there is no symplectic resolution. We thereby reduce the classification problem to finitely many open cases. We furthermore prove non-existence of a symplectic resolution for one exceptional group, leaving $$39+9=48$$ 39 + 9 = 48 open cases in total. We do not expect any of the remaining cases to admit a symplectic resolution.

Algebraic cycles and intersections of a quadric and a cubic

Let Y be a smooth complete intersection of a quadric and a cubic in ℙ n {\mathbb{P}^{n}} , with n even. We show that Y has a multiplicative Chow–Künneth decomposition, in the sense of Shen–Vial. As a consequence, the Chow ring of (powers of) Y displays K3-like behavior. As a by-product of the argument, we also establish a multiplicative Chow–Künneth decomposition for the resolution of singularities of a general nodal cubic hypersurface of even dimension.

POWER GEOMETRY IN LOCAL RESOLUTION OF SINGULARITIES OF AN ALGEBRAIC CURVE

The main goal of this work is to provide a consistent set of general-purpose algorithms for analyzing singularities applicable to all types of equations. We present the main ideas and algorithms of power geometry and give an overview of some of its applications. We also present a procedure that allows us to distinguish all branches of a spatial curve near a singular point and calculate the parametric appearance of these branches with any degree of accuracy. For a specific case, we show how this algorithm works.