Positivity in Algebraic Geometry I

Abstract. In this paper, we associate an invariant α x (L) to an algebraic point x on an algebraic variety X with an ample line bundle L. The invariant α measures how well x can be approximated by rational points on X, with respect to the height function associated to L. We show that this invariant is closely related to the Seshadri constant ǫ x (L) measuring local positivity of L at x, and in particular that Roth's theorem on P 1 generalizes as an inequality between these two invariants valid for arbitrary projective varieties.

show abstract

“…Recall that by [11,Proposition 5.1.9] for any irreducible subvariety V ⊆ X of positive dimension passing through x we have the inequality…”

mentioning

confidence: 99%

Seshadri constants, diophantine approximation, and Roth’s theorem for arbitrary varieties

McKinnon

Roth

2014

Invent. math.

View full text Add to dashboard Cite

show abstract

“…Furthermore, it is contained in the dual of the pseudoeffective cone Eff(X) ⊂ N 1 (X) R (cf. [15], Remark 2.2.28). In the case of surfaces the dual of Eff(X) is the nef cone of the surface.…”

Section: Remarksmentioning

confidence: 93%

Quotient categories, stability conditions, and birational geometry

Meinhardt

Partsch

2014

Geom Dedicata

View full text Add to dashboard Cite

Abstract. This article deals with the quotient category of the category of coherent sheaves on an irreducible smooth projective variety by the full subcategory of sheaves supported in codimension greater than c. It turns out that this category has homological dimension c. As an application of this, we will describe the space of stability conditions on its derived category in the case c=1. Moreover, we describe all exact equivalences between these quotient categories in this particular case which is closely related to classification problems in birational geometry.

show abstract

“…Furthermore, we can assume that there is a morphism f : X −→ Y giving the Iitaka fibration for K X + with Y a smooth projective variety of dimension m [Lazarsfeld 2004, 2.1.C]. For F the general fiber of f , we have that κ(K F + F ) = 0.…”

Section: Effectiveness Of the Iitaka Fibrationmentioning

confidence: 99%