Segre class computation and practical applications
TLDR
In this paper, the Chow group of projective toric toric varieties has been used to compute intersection products in projective varieties, to determine algebraic multiplicity without working in local rings, and to test pairwise containment of subvarieties of a toric variety.Abstract:
Let $X \subset Y$ be closed (possibly singular) subschemes of a smooth projective toric variety $T$. We show how to compute the Segre class $s(X,Y)$ as a class in the Chow group of $T$. Building on this, we give effective methods to compute intersection products in projective varieties, to determine algebraic multiplicity without working in local rings, and to test pairwise containment of subvarieties of $T$. Our methods may be implemented without using Groebner bases; in particular any algorithm to compute the number of solutions of a zero-dimensional polynomial system may be used.read more
Citations
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The Maximum Likelihood Degree of Linear Spaces of Symmetric Matrices
TL;DR: In this paper, the authors studied multivariate Gaussian models that are described by linear conditions on the concentration matrix and computed the maximum likelihood (ML) degrees of these models, i.e., the critical points of the likelihood function over a linear space of symmetric matrices.
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Conormal Spaces and Whitney Stratifications
TL;DR: In this paper , a new algorithm for computing Whitney stratifications of complex projective varieties is described, which improves upon the existing state-of-the-art by several orders of magnitude, even for relatively small input varieties.
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Maximum Likelihood Estimation of Toric Fano Varieties
TL;DR: In this paper, the maximum likelihood estimation problem for several classes of toric Fano models was studied, and it was shown that the ML degree is equal to the degree of the surface in every case except for the quintic del Pezzo surface with two ordinary double points.
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Maximum likelihood estimation of toric Fano varieties
TL;DR: In this article, the maximum likelihood estimation problem for several classes of toric Fano models was studied, and it was shown that the ML degree is equal to the degree of the surface in every case except for the quintic del Pezzo surface.
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Segre classes and invariants of singular varieties
TL;DR: In this paper, a survey of applications of Segre classes to the definition and study of invariants of singular spaces is presented, focusing on several numerical invariants, on different notions of characteristic classes for singular varieties, and on classes of Le cycles.
References
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Book
Combinatorial Commutative Algebra
Ezra Miller,Bernd Sturmfels +1 more
TL;DR: In this paper, the authors present a set of monomial ideals for three-dimensional staircases and cellular resolutions, including two-dimensional lattice ideals, and a threedimensional staircase with cellular resolutions.
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Algebraic Geometry: A First Course
TL;DR: In this article, the authors introduce the notion of Tangent Spaces to Grassmannians and describe the relationship between them and regular functions and maps. But they do not discuss their application in the context of dimension computations.
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The geometry of toric varieties
TL;DR: Affine toric varieties have been studied in this article, where the definition of an affine Toric variety and its properties have been discussed, including cones, lattices, and semigroups.
Journal ArticleDOI
Algorithm 795: PHCpack: a general-purpose solver for polynomial systems by homotopy continuation
TL;DR: The structure and design of the software package PHC is described, which features great variety of root-counting methods among its tools and is ensured by the gnu-ada compiler.