Real fibered morphisms and Ulrich sheaves

Kummer, Mario; Shamovich, Eli

doi:10.1090/jag/735

Cited by 13 publications

(29 citation statements)

References 60 publications

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“…Theorem (Theorem 4.5 [26]). Under the conditions above, there is an Ulrich module M over S that is isomorphic to R k as R-module, and whose annihilator is I(X).…”

Section: Ulrich Modules and The Entropic Discriminantmentioning

confidence: 99%

“…The proof uses the equivalence, recently developed in [26], between Livšic-type determinantal representations and certain types of Ulrich modules. An Ulrich module of a polynomial ring S = R[x 1 , .…”

Section: Ulrich Modules and The Entropic Discriminantmentioning

confidence: 99%

“…, v n ∈ R n be a basis for L ⊥ . To use the construction described in [26], we need to do a change of basis on R n to write the projection from L ⊥ as a coordinate projection. Specifically, for each i = 1, .…”

Section: Ulrich Modules and The Entropic Discriminantmentioning

confidence: 99%

“…Section 5 discusses relations to the Hadamard product of linear spaces, as studied in [4]. The results of [26] and this paper imply that there exist rank-one Ulrich sheaves on reciprocal linear spaces. In Section 6, we use this to prove the conjecture of [38] that the entropic discriminant is a sum of squares.…”

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confidence: 92%

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The Chow Form of a Reciprocal Linear Space

Kummer¹,

Vinzant²

2019

Michigan Math. J.

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A reciprocal linear space is the image of a linear space under coordinate-wise inversion. These fundamental varieties describe the analytic centers of hyperplane arrangements and appear as part of the defining equations of the central path of a linear program. Their structure is controlled by an underlying matroid. This provides a large family of hyperbolic varieties, recently introduced by Shamovich and Vinnikov. Here we give a definite determinantal representation to the Chow form of a reciprocal linear space. One consequence is the existence of symmetric rank-one Ulrich sheaves on reciprocal linear spaces. Another is a representation of the entropic discriminant as a sum of squares. For generic linear spaces, the determinantal formulas obtained are closely related to the Laplacian of the complete graph and generalizations to simplicial matroids. This raises interesting questions about the combinatorics of hyperbolic varieties and connections with the positive Grassmannian.

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“…Theorem (Theorem 4.5 [26]). Under the conditions above, there is an Ulrich module M over S that is isomorphic to R k as R-module, and whose annihilator is I(X).…”

Section: Ulrich Modules and The Entropic Discriminantmentioning

confidence: 99%

Section: Ulrich Modules and The Entropic Discriminantmentioning

confidence: 99%

Section: Ulrich Modules and The Entropic Discriminantmentioning

confidence: 99%

mentioning

confidence: 92%

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The Chow Form of a Reciprocal Linear Space

Kummer¹,

Vinzant²

2019

Michigan Math. J.

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“…We outline the proof of the above theorem for completeness, referring the reader to [KS15] for details. Since it is a local statement, we may assume that both X and Y are real open neighbourhoods of 0 in C n .…”

Section: Properties Of the Logarithmic Gauß Mapmentioning

confidence: 99%

Nonexistence of torically maximal hypersurfaces

Brugallé

Mikhalkin

Risler

et al. 2018

St. Petersburg Math. J.

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Real-Fibered Morphisms of del Pezzo Surfaces and Conic Bundles

Kummer

Texier

Manzaroli

2022

Discrete Comput Geom

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It goes back to Ahlfors that a real algebraic curve admits a real-fibered morphism to the projective line if and only if the real part of the curve disconnects its complex part. Inspired by this result, we are interested in characterising real algebraic varieties of dimension n admitting real-fibered morphisms to the n-dimensional projective space. We present a criterion to classify real-fibered morphisms that arise as finite surjective linear projections from an embedded variety which relies on topological linking numbers. We address special attention to real algebraic surfaces. We classify all real-fibered morphisms from real del Pezzo surfaces to the projective plane and determine which such morphisms arise as the composition of a projective embedding with a linear projection. Furthermore, we give some insights in the case of real conic bundles.

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Real fibered morphisms and Ulrich sheaves

Cited by 13 publications

References 60 publications

The Chow Form of a Reciprocal Linear Space

The Chow Form of a Reciprocal Linear Space

Nonexistence of torically maximal hypersurfaces

Real-Fibered Morphisms of del Pezzo Surfaces and Conic Bundles

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