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Henning Lohne

Publications and source records attributed to Henning Lohne.

4 recordsLinked to original sources
arXiv2020

Polarizations of powers of graded maximal ideals

We give a complete combinatorial characterization of all possible polarizations of powers of the graded maximal ideal $(x_1,x_2,\cdots,x_m)^n$ of a polynomial ring in $m$ variables. We also give a combinatorial description of the Alexander duals of such polarizations. In the three variable case $m=3$ and also in the power two case $n=2$ the descriptions are easily visualized and we show that every polarization defines a (shellable) simplicial ball. We give conjectures relating to topological properties and to algebraic geometry, in particular that any polarization of an Artinian monomial ideal defines a simplicial ball.

math.AC↗
arXiv2013

The many polarizations of powers of maximal ideals

In this paper, we study different polarizations of powers of the maximal ideal m^d, and polarizations of its related square-free version I_d. For n = 3, we show that every minimal free cellular resolution of m^d comes from a certain polarization of the ideal m^d. When I is a square-free ideal, we show that the Alexander dual of any polarization of I is a polarization of the Alexander dual ideal of I. We apply this theorem and study different polarizations of I_d and its Alexander dual I_{n-d+1} simultaneously. We do this by giving a combinatorial description of the polarizations, which has a natural duality. We study the case of d = 2 and d = n-1 in more detail. Here, we show that there is a one- to-one correspondence between spanning trees of the complete graph K_n and the maximal polarizations of these ideals.

math.AC↗
arXiv2013

Deformations of polarizations of powers of the maximal ideal

In this paper, we study the two natural polarizations, namely the standard polarization and the box polarization, of the d-th power of the maximal ideal in a polynomial ring. We show that these polarizations correspond to smooth points in the Hilbert scheme, and we calculate the dimension of their component which shows that they lie on different components. When d=2, we show that all maximal polarizations are smooth points, and we give a simple method for calculating the dimension of their component.

math.AC↗