Hadamard products of linear spaces

Bocci, Cristiano; Carlini, Enrico; Kileel, Joe

doi:10.1016/j.jalgebra.2015.10.008

Cited by 41 publications

(88 citation statements)

References 29 publications

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“…⋆ X ℓ is the sum of the dimensions, the degree is the product of the degrees multiplied by a multi-binomial coefficient depending on the dimensions and the Hilbert function is the product of the Hilbert functions. These degree and dimension formulas generalize the ones in [BCK,Theorem 6.8] which are only given for linear spaces. We also prove that, if the varieties X i are smooth, then their Hadamard product is smooth.…”

Section: Introductionsupporting

confidence: 66%

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On the Hadamard product of degenerate subvarieties

et al. 2020

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We consider generic degenerate subvarieties Xi ⊂ P n . We determine an integer N , depending on the varieties, and for n ≥ N we compute dimension and degree formulas for the Hadamard product of the varieties Xi. Moreover, if the varieties Xi are smooth, their Hadamard product is smooth too. For n < N , if the Xi are generically di-parameterized, the dimension and degree formulas still hold. However, the Hadamard product can be singular and we give a lower bound for the dimension of the singular locus.

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Section: Introductionsupporting

confidence: 66%

“…The Hadamard product of varieties is also related to tropical geometry, for instance the tropicalization of the Hadamard product of two varieties is the Minkowski sum of the tropicalizations of the two varieties (see [BCK,Proposition 5.1], [FOW], [MS]).…”

Section: Introductionmentioning

confidence: 99%

On the Hadamard product of degenerate subvarieties

et al. 2020

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“…We round out this section by explaining the connection to [3]. We first need the following variant of the Hadamard product.…”

Section: Background Resultsmentioning

confidence: 99%

“…In [3] the authors show how to construct zero dimensional star configurations by using a sufficiently general line ℓ ⊂ P n . In particular, after one fixes points P 1 , .…”

Section: Toward Explicit Constructions Of Hadamard Star Configurationmentioning

confidence: 99%

Hadamard star configurations

Carlini¹,

Catalisano²,

Guardo³

et al. 2019

Rocky Mountain J. Math.

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Bocci, Carlini, and Kileel have shown that the square-free Hadamard product of a finite set of points Z that all lie on a line ℓ in P n produces a star configuration of codimension n. In this paper we introduce a construction using the Hadamard product to construct star configurations of codimension c. In the case that c = n = 2, our construction produces the star configurations of Bocci, Carlini, and Kileel. We will call any star configuration that can be constructed using our approach a Hadamard star configuration. Our main result is a classification of Hadamard star configurations.

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“…× P 1 ⊂ P 2 n −1 with itself. Further study in [BCK16], [FOW17], [CCFL] made progress towards understanding Hadamard products. It seems natural to look at the r-th Hadamard power X r := X .…”

Section: Introductionmentioning

confidence: 99%

Coordinate-wise powers of algebraic varieties

2020

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We introduce and study coordinate-wise powers of subvarieties of P n , i.e. varieties arising from raising all points in a given subvariety of P n to the r-th power, coordinate by coordinate. This corresponds to studying the image of a subvariety of P n under the quotient of P n by the action of the finite group Z n+1 r . We determine the degree of coordinate-wise powers and study their defining equations, in particular for hypersurfaces and linear spaces. Applying these results, we compute the degree of the variety of orthostochastic matrices and determine iterated dual and reciprocal varieties of power sum hypersurfaces. We also establish a link between coordinate-wise squares of linear spaces and the study of real symmetric matrices with a degenerate eigenspectrum.

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Hadamard products of linear spaces

Abstract: We describe properties of Hadamard products of algebraic varieties. We show any Hadamard power of a line is a linear space, and we construct star configurations from products of collinear points. Tropical geometry is used to find the degree of Hadamard products of other linear spaces

Cited by 41 publications

References 29 publications

On the Hadamard product of degenerate subvarieties

On the Hadamard product of degenerate subvarieties

Hadamard star configurations

Coordinate-wise powers of algebraic varieties

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