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It is known that, unlike the one dimensional case it is not possible to find an upper bound for the zeros of an entire map from$C$^{$n$}to$C$^{$n$},$n$≥2, in terms of the growth of the map. However, if we only consider the “non-degenerate” zeros, that is, the zeros where the jacobian is not “too small”, it becomes possible. We give a new proof of this fact.

We study the geometry of Hilbert schemes of points on abelian surfaces and Beauville’s generalized Kummer varieties in positive characteristics. The main result is that, in characteristic two, the addition map from the Hilbert scheme of two points to the abelian surface is a quasifibration such that all fibers are nonsmooth. In particular, the corresponding generalized Kummer surface is nonsmooth, and minimally elliptic singularities occur in the supersingular case. We unravel the structure of the singularities in dependence of$p$-rank and$a$-number of the abelian surface. To do so, we establish a McKay Correspondence for Artin’s wild involutions on surfaces. Along the line, we find examples of canonical singularities that are not rational singularities.

In this paper we use a homological approach to obtain upper bounds for a few homological invariants of FI_G-modules V. These upper bounds are expressed in terms of the generating degree and torsion degree, which measure the top and socle of V under actions of non-invertible morphisms in the category respectively.