On k-Jet Ampleness

“…We note here that if X is a smooth algebraic variety and L is a very-ample line bundle then L N is k-jet spanned for all N k, see [BS93]. Bounds on N for when a multiple L N of (just) an ample line bundle L is k-jet spanned have been investigated for many classes of varieties, as K3 surfaces [BDRSz00] or Abelian varieties [BSz97].…”

Section: 2mentioning

confidence: 99%

“…Bounds on N for when a multiple L N of (just) an ample line bundle L is k-jet spanned have been investigated for many classes of varieties, as K3 surfaces [BDRSz00] or Abelian varieties [BSz97]. More generally k-jet spanned embeddings have been studied for many classes of varieties, see for example [BS93,Te98,BDRS98].…”

Section: 2mentioning

confidence: 99%

A note on higher-order Gauss maps

Rocco¹,

Jabbusch²,

Lundman³

2017

Michigan Math. J.

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ABSTRACT. We study Gauss maps of order k, associated to a projective variety X embedded in projective space via a line bundle L. We show that if X is a smooth, complete complex variety and L is a k-jet spanned line bundle on X, with k 1, then the Gauss map of order k has finite fibers, unless X = P n is embedded by the Veronese embedding of order k. In the case where X is a toric variety, we give a combinatorial description of the Gauss maps of order k, its image and the general fibers.

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“…The notion of k-jet ampleness generalises the notion of very ampleness and k-very ampleness (see [8,Proposition 2.2]). We recall the definition of k-very ampleness, as we mention this notion it the proof of the main theorem: Definition 2.2.…”

Section: Preliminariesmentioning

confidence: 99%

“…The concepts of higher order embeddings: k-spandness, k-very ampleness, and k-jet ampleness were introduced and studied in a series of papers by Beltrametti, Francia, and Sommese, see [6][7][8]. The last notion is of our main interest in the present work.…”

Section: Introductionmentioning

confidence: 99%