Titlebook: Hadamard Products of Projective Varieties; Cristiano Bocci,Enrico Carlini Book 2024 The Editor(s) (if applicable) and The Author(s), under

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Hilbert Functions,In this chapter, we will describe some connections among the Hilbert functions of ., . and .. However, a complete description of these connections is not known yet. In this chapter, we focus our attention on linear spaces, collinear points, degenerate varieties, and binomial varieties.

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Star Configurations,In this chapter, we give a relevant application of the Hadamard powers of a line: Theorem 9.1shows how to naturally construct star configurations using Hadamard products. The interest in star configurations has recently increased because of their extremal behaviour with respect to many open problems such as, for example, symbolic powers of ideals.

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Open Questions,In this chapter, we introduce some open problems about Hadamard products of projective varieties and related topics. Since this is an active and young area of research, we hope that this chapter will give a chance to the interested reader to actively get involved in the subject.

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Volodymyr Pugachov,Nikolay Pugachovrd products, and then we describe the main tools, such as Hadamard transformations, and some basic results, such as Hadamard–Terracini Lemma, that we will use in the whole book. In this chapter, we also consider the Hadamard product of ideals proving some relevant results related to Gr?bner bases.

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https://doi.org/10.1007/978-3-319-46388-9tiplication of linear spaces by a point. Then, we study Hadamard products, and powers, of lines. Successively, we consider Hadamard products of linear spaces. Using Tropical Geometry, we are able to give some insight on the degree and the dimension of such Hadamard products. The last section introdu

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Delineating the Boundaries of Discourse,near spaces: Points can have many zero coordinates, and linear spaces can intersect coordinate hyperplanes in dimension greater than the expected one. These facts force us to study all possible pathological behaviours that can occur when computing the Hadamard product of such varieties.

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David F. Lomax,P. T. G. Gutmannon the coordinates of the points forming the star configuration and on the equations of the hyperplanes involved. Thus, the question if other interesting geometrical objects can be obtained via Hadamard products naturally arises. In Sect. ., for example, we used Hadamard products to construct planar

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The Political and Economic Background,to do this, we used radical ideals associated to the varieties, but Definition . allows us to use different families of ideals, such as monomial ideals, square-free binomial ideals, and edge ideals. The possibility to use these different families of ideals opens up new and challenging research oppor

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Linear Spaces,tiplication of linear spaces by a point. Then, we study Hadamard products, and powers, of lines. Successively, we consider Hadamard products of linear spaces. Using Tropical Geometry, we are able to give some insight on the degree and the dimension of such Hadamard products. The last section introdu