Monads on projective varieties

Monads on Projective Space

Representations of quivers on abelian categories and monads on projective varieties

We consider representations of quivers in arbitrary categories and twisted representations of quivers in arbitrary tensor categories. We show that if A is an abelian category, then the category of representations of a quiver in A is also abelian, and that the category of twisted linear representations of a quiver is equivalent to the category of linear (untwisted) representations of a different...

Lines on Projective Varieties

I prove two theorems: Let X ⊂ P be a hypersurface and let x ∈ X be a general point. If the set of lines having contact to order k with X at x is of dimension greater than expected, then the lines having contact to order k are actually contained in X. A variety X is said to be covered by lines if there exist a finite number of lines in X passing through a general point. Let X ⊂ P be a variety co...

Diophantine Inequalities on Projective Varieties

then the set of solutions of (1.1) lies in the union of finitely many proper linear subspaces of P. We give an equivalent formulation on which we shall focus in this paper. Let {l0, . . . , lN} be the union of the sets {l0v, . . . , lnv} (v ∈ S). Define the map φ : P → P by y 7→ ( l0(y) : · · · : lN(y) ) . Put X := φ(P); then X is a linear subvariety of P of dimension n defined over K. Write xi...

Quantitative Diophantine Approximations on Projective Varieties

In this note we study some aspects of a paper of G. Faltings and G. Wüstholz [7]. In particular, we work out a quantitative version of their main theorem. Our result even gives an explicit presentation of the structure for the exceptional subspaces of [24]. For a generic projective variety we obtain double exponential estimates. However, knowing more about the geometry of the given variety one ...