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We seek to determine a real algebraic variety from a fixed finite subset of points. Existing methods are studied and new methods are developed. Our focus lies on aspects of topology and algebraic geometry, such as dimension and defining polynomials. All algorithms are tested on a range of datasets and made available in a Julia package.
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In this article we review the question of constructing geometric quotients of actions of linear algebraic groups on irreducible varieties over algebraically closed fields of characteristic zero, in the spirit of Mumfords geometric invariant theory (GIT). The article surveys some recent work on geometric invariant theory and quotients of varieties by linear algebraic group actions, as well as background material on linear algebraic groups, Mumfords GIT and some of the challenges that the non-reductive setting presents. The earlier work of two of the authors in the setting of unipotent group actions is extended to deal with actions of any linear algebraic group. Given the data of a linearisation for an action of a linear algebraic group H on an irreducible variety $X$, an open subset of stable points $X^s$ is defined which admits a geometric quotient variety $X^s/H$. We construct projective completions of the quotient $X^s/H$ by considering a suitable extension of the group action to an action of a reductive group on a reductive envelope and using Mumfords GIT. In good cases one can also compute the stable locus $X^s$ in terms of stability (in the sense of Mumford for reductive groups) for the reductive envelope.
Using the work of Guillen and Navarro Aznar we associate to each real algebraic variety a filtered chain complex, the weight complex, which is well-defined up to filtered quasi-isomorphism, and which induces on Borel-Moore homology with Z/2 coefficients an analog of the weight filtration for complex algebraic varieties.
Let X be a smooth algebraic variety over a field of characteristic 0. We introduce the notion of twisted associative (resp. Poisson) deformation of the structure sheaf O_X. These are stack-li
Let $ Y subseteq Bbb P^N $ be a possibly singular projective variety, defined over the field of complex numbers. Let $X$ be the intersection of $Y$ with $h$ general hypersurfaces of sufficiently large degrees. Let $d>0$ be an integer, and assume that $dim Y=n+h$ and $ dim Y_{sing} le min{ d+h-1 , n-1 } $. Let $Z$ be an algebraic cycle on $Y$ of dimension $d+h$, whose homology class in $H_{2(d+h)}(Y; Bbb Q)$ is non-zero. In the present paper we prove that the restriction of $Z$ to $X$ is not algebraically equivalent to zero. This is a generalization to the singular case of a result due to Nori in the case $Y$ is smooth. As an application we provide explicit examples of singular varieties for which homological equivalence is different from the algebraic one.
We extend the Altmann-Hausen presentation of normal affine algebraic C-varieties endowed with effective torus actions to the real setting. In particular, we focus on actions of quasi-split real tori, in which case we obtain a simpler presentation.
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