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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.
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 prove
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 vari
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 t