We introduce a notion of ampleness for subschemes of any codimension using the theory of q-ample line bundles. We also investigate certain geometric properties satisfied by ample subvarieties, e.g. the Lefschetz hyperplane theorems and numerical positivity. Using these properties, we also construct a counterexample to the converse of the Andreotti-Grauert vanishing theorem.

Using geodesic length functions, we define a natural family of real codimension 1 subvarieties of Teichmüller space, namely the subsets where the lengths of two distinct simple closed geodesics are of equal length. We investigate the point set topology of the union of all such hypersurfaces using elementary methods. Finally, this analysis is applied to investigate the nature of the Markoff conj...

Let X be a codimension 1 subvariety of dimension > 1 of a variety of minimal degree Y . If X is subcanonical with Gorenstein canonical singularities admitting a crepant resolution, then X is Arithmetically Gorenstein and we characterise such subvarieties X of Y via apolarity as those whose apolar hypersurfaces are Fermat.

We give some experimental data of Gorenstein liaison, working with points in P and curves in P , to see how far the familiar situation of liaison, biliaison, and Rao modules of curves in P will extend to subvarieties of codimension 3 in higher P . AMS classification numbers: 14H50, 14M07

We study holomorphic foliations of arbitrary codimension in smooth complete toric varieties. show that split are stable if some good behaviour their singular set is provided. As an application these results, we exhibit irreducible components the space arise as pullbacks special T-invariant subvarieties.

Two birational projective varieties in $$\mathbb {P}^n$$ are Cremona Equivalent if there is a modification of mapping one onto the other. The minimal degree $$X\subset \mathbb integer among all degrees that to X. Equivalence and well understood for subvarieties codimension at least 2 while both general very subtle questions divisors. In this note, I compute quartic surfaces {P}^3$$ . This allow...

We study compactifications of subvarieties of algebraic tori defined by imposing a sufficiently fine polyhedral structure on their non-archimedean amoebas. These compactifications have many nice properties, for example any k boundary divisors intersect in codimension k. We consider some examples including M0,n ⊂ M0,n (and more generally log canonical models of complements of hyperplane arrangem...

Abstract. We introduce a method to determine if n-dimensional smooth subvarieties of an ambient space of dimension at most 2n − 2 inherit the Picard group from the ambient space (as it happens when the ambient space is a projective space, according to results of Barth and Larsen). As an application, we give an affirmative answer (up to some mild natural numerical conditions) when the ambient sp...

Let X be a smooth, complex Fano variety, and delta(X) its Lefschetz defect. It is known that if at least 4, then isomorphic to product SxT, where dim T=dim X-2. In this paper we prove structure theorem for the case delta(X)=3. We show there exists smooth variety T with X-2 such obtained from two possible explicit constructions; in both cases P^2-bundle Z over blow-up of along three pairwise dis...