Abstract. A complex frame is a collection of vectors that span C M and define measurements, called intensity measurements, on vectors in C M . In purely mathematical terms, the problem of phase retrieval is to recover a complex vector from its intensity measurements, namely the modulus of its inner product with these frame vectors. We show that any vector is uniquely determined (up to a global phase factor) from 4M − 4 generic measurements. To prove this, we identify the set of frames defining non-injective measurements with the projection of a real variety and bound its dimension.
We show that a torus-equivariant vector bundle on a complete toric variety is nef or ample if and only if its restriction to every invariant curve is nef or ample, respectively. Furthermore, we show that nef toric vector bundles have a nonvanishing global section at every point and deduce that the underlying vector bundle is trivial if and only if its restriction to every invariant curve is trivial. We apply our methods and results to study, in particular, the vector bundles M L that arise as the kernel of the evaluation map H 0 (X, L) ⊗ O X → L, for ample line bundles L. We give examples of twists of such bundles that are ample but not globally generated.Résumé. Nous prouvons qu'un fibré vectoriel equivariant sur une variété torique complète est nef ou ample si et seulement si sa restriction à chaque courbe invariante est nef ou ample, respectivement. Nous montrons également qu'étant donne un fibré vectoriel torique nef E et un point x ∈ X, il existe une section de E non-nulle en x; on déduit de cela que E est trivial si et seulement si sa restriction à chaque courbe invariante est triviale. Nous appliquons ces résultats et méthodes pour étudier en particulier les fibrés vectoriels M L , définis en tant que noyau des applications d'évaluation H 0 (X, L) ⊗ O X → L, ou L est un fibré en droites ample. Finalement, nous donnons des exemples des fibrés vectoriels toriques qui sont amples mais non engendrés par leur sections globales.
Using multigraded Castelnuovo-Mumford regularity, we study the equations defining a projective embedding of a variety X. Given globally generated line bundles B 1 , . . . , B on X and m 1 , . . . , m ∈ N, consider the line bundle L := B m 1 1 ⊗· · · ⊗B m . We give conditions on the m i which guarantee that the ideal of X in P(H 0 (X, L) * ) is generated by quadrics and that the first p syzygies are linear. This yields new results on the syzygies of toric varieties and the normality of polytopes.
We establish an explicit bijection between the toric degenerations of the Grassmannian Gr(2, n) arising from maximal cones in tropical Grassmannians and the ones coming from plabic graphs corresponding to Gr(2, n). We show that a similar statement does not hold for Gr (3, 6).
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