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Register a new userRamblings on the freeness of affine hypersurfaces
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In this note we look at the freeness for complex affine hypersurfaces. If $X subset mathbb{C}^n$ is such a hypersurface, and $D$ denotes the associated projective hypersurface, obtained by taking the closure of $X$ in $mathbb{P}^n$, then we relate first the Jacobian syzygies of $D$ and those of $X$. Then we introduce two types of freeness for an affine hypersurface $X$, and prove various relations between them and the freeness of the projective hypersurface $D$. We write down a proof of the folklore result saying that an affine hypersurface is free if and only if all of its singularities are free, in the sense of K. Saitos definition in the local setting. In particular, smooth affine hypersurfaces and affine plane curves are always free. Some other results, involving global Tjurina numbers and minimal degrees of non trivial syzygies are also explored.
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Here we prove that the Hilbert-Kunz mulitiplicity of a quadric hypersurface of dimension $d$ and odd characteristic $pgeq 2d-4$ is bounded below by $1+m_d$, where $m_d$ is the $d^{th}$ coefficient in the expansion of $mbox{sec}+mbox{tan}$. This proves a part of the long standing conjecture of Watanabe-Yoshida. We also give an upper bound on the HK multiplicity of such a hypersurface. We approach the question using the HK density function and the classification of ACM bundles on the smooth quadrics via matrix factorizations.
By way of Ziegler restrictions we study the relation between nearly free plane arrangements and combinatorics and we give a Yoshinaga-type criterion for plus-one generated plane arrangements.
We study the critical points of the likelihood function over the Fermat hypersurface. This problem is related to one of the main problems in statistical optimization: maximum likelihood estimation. The number of critical points over a projective variety is a topological invariant of the variety and is called maximum likelihood degree. We provide closed formulas for the maximum likelihood degree of any Fermat curve in the projective plane and of Fermat hypersurfaces of degree 2 in any projective space. Algorithmic methods to compute the ML degree of a generic Fermat hypersurface are developed throughout the paper. Such algorithms heavily exploit the symmetries of the varieties we are considering. A computational comparison of the different methods and a list of the maximum likelihood degrees of several Fermat hypersurfaces are available in the last section.
We classify the subvarieties of infinite dimensional affine space that are stable under the infinite symmetric group. We determine the defining equations and point sets of these varieties as well as the containments between them.
In this paper we study tensor products of affine abelian group schemes over a perfect field $k.$ We first prove that the tensor product $G_1 otimes G_2$ of two affine abelian group schemes $G_1,G_2$ over a perfect field $k$ exists. We then describe the multiplicative and unipotent part of the group scheme $G_1 otimes G_2$. The multiplicative part is described in terms of Galois modules over the absolute Galois group of $k.$ We describe the unipotent part of $G_1 otimes G_2$ explicitly, using Dieudonne theory in positive characteristic. We relate these constructions to previously studied tensor products of formal group schemes.
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