On the ampleness of invertible sheaves in complete projective toric varieties

Ewald, Günter; Wessels, Uwe

doi:10.1007/bf03323286

Cited by 39 publications

(29 citation statements)

References 3 publications

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“…Recently, Haase, Nill, Pfaffenholz and Santos [18] were able to prove this for arbitrary toric surfaces. Moreover it is well known [10,26,3] that for an ample line bundle L on a possibly singular toric variety of dimension n, the multiplication map…”

Section: Vector Bundle Whose Behavior Is Closely Related To the Geomementioning

confidence: 99%

Positivity properties of toric vector bundles

Hering¹,

Mustaţă²,

Payne³

2010

Ann. inst. Fourier

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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.

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Section: Vector Bundle Whose Behavior Is Closely Related To the Geomementioning

confidence: 99%

Positivity properties of toric vector bundles

Hering¹,

Mustaţă²,

Payne³

2010

Ann. inst. Fourier

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“…3.3 is essentially equivalent to the results of Ewald and Wessels [13] and Liu, Trotter, and Ziegler [21] which, however, have been derived by different methods. 3.3 is essentially equivalent to the results of Ewald and Wessels [13] and Liu, Trotter, and Ziegler [21] which, however, have been derived by different methods.…”

Section: 4 Lei P Be a Lattice N-polytope Then The Normalization Ofmentioning

confidence: 64%

Normal polytopes, triangulations, and Koszul algebras.

Bruns¹,

Gubeladze²,

NguyenKhac³

1997

Journal Für Die Reine Und Angewandte Mathematik (Crelles Journal)

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“…So it makes sense to apply the technique in this setting. In [4], Ewald and Wessels prove that if D is an ample divisor on a toric variety of dimension n, then (n − 1)D is very ample and satisfies N 0 . Bruns, Gubeladze and Trung [2] give another proof and also show that nD satisfies property N 1 .…”

Section: Green's Theorem and Hyperplane Sectionsmentioning

confidence: 99%

“…So at first glance the bound above seems useless. However, when n = 2 the term n(n − 2)vol(P ) vanishes, and by [4] the divisor associated to a lattice polygon P satisfies N 0 . So we obtain:…”

Section: Applicationsmentioning

confidence: 99%

Lattice polygons and Green’s theorem

Schenck

2004

Proc. Amer. Math. Soc.

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Abstract. Associated to an n-dimensional integral convex polytope P is a toric variety X and divisor D, such that the integral points of P represent H 0 (O X (D)). We study the free resolution of the homogeneous coordinate ring m∈Z H 0 (mD) as a module over Sym(H 0 (O X (D))). It turns out that a simple application of Green's theorem yields good bounds for the linear syzygies of a projective toric surface. In particular, for a planar polytope, D satisfies Green's condition Np if ∂P contains at least p + 3 lattice points. Green's theorem and hyperplane sectionsFor a curve C of genus g, a divisor D of degree d ≥ 2g+1 is very ample, so gives an embedding of C into projective space. In fact, when d ≥ 2g+1, work of Castelnuovo, Mattuck and Mumford shows that the embedding is projectively normal, which means that, results of Fujita and St. Donat show that the homogeneous ideal of I C is generated by quadrics. Let F • be a minimal free resolution of R over S. A very ample divisor is said to satisfy property N p if F 0 = S and F q S(−q − 1) for all q ∈ {1, . . . , p}. Thus, N 0 means projectively normal, N 1 means that the homogeneous ideal is generated by quadrics, N 2 means that the minimal syzygies on the quadrics are linear, and so on. In [7], Green used Koszul cohomology to give a beautiful generalization of the classical results above:In this brief note, we investigate the N p property for toric varieties. For any divisor D and variety X such that R is arithmetically Cohen-Macaulay, it is natural to slice with hyperplanes until X has been reduced to a curve, and then apply Green's theorem. Results of Hochster [8] show that projectively normal toric varieties are always arithmetically Cohen-Macaulay. So it makes sense to apply the technique in this setting. In [4], Ewald and Wessels prove that if D is an ample divisor on a toric variety of dimension n, then (n − 1)D is very ample and satisfies N 0 . Bruns, Gubeladze and Trung [2] give another proof and also show that nD satisfies property N 1 . While it is often difficult to determine if a given divisor satisfies N 0 , for a lattice polygon P and corresponding divisor on a toric surface, the property N 0 holds "for free".

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On the ampleness of invertible sheaves in complete projective toric varieties

Cited by 39 publications

References 3 publications

Positivity properties of toric vector bundles

Positivity properties of toric vector bundles

Normal polytopes, triangulations, and Koszul algebras.

Lattice polygons and Green’s theorem

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