We study higher syzygies of a ruled surface X over a curve of genus g with the numerical invariant e. Let L ∈ PicX be a line bundle in the numerical class of aC0 + bf . We prove that for 0 ≤ e ≤ g − 2, L satisfies Property Np if a ≥ p + 2 and b − ae ≥ 3g − e + p and for e ≥ g − 1, L satisfies Property Np if a ≥ p + 2 and b − ae ≥ 2g + 1 + p. By using these facts, we obtain Mukai type results. F...

We study ruled orders. These arise naturally in the Mori program for orders on projective surfaces and morally speaking are orders on a ruled surface ramified on a bisection and possibly some fibres. We describe fibres of a ruled order and show they are in some sense rational. We also determine the Hilbert scheme of rational curves and hence the corresponding non-commutative Mori contraction. T...

In this article we we continue the study of property Np of irrational ruled surfaces begun in [12]. Let X be a ruled surface over a curve of genus g ≥ 1 with a minimal section C0 and the numerical invariant e. When X is an elliptic ruled surface with e = −1, there is an elliptic curve E ⊂ X such that E ≡ 2C0 − f . And we prove that if L ∈ PicX is in the numerical class of aC0 + bf and satisfies...

Let X be a non-singular ruled surface over an algebraically closed field of characteristic zero. There is a non-trivial surjective endomorphism f : X → X if and only if X is (1) a toric surface, (2) a relatively minimal elliptic ruled surface, or (3) a relatively minimal ruled surface of irregularity greater than one which turns to be the product of P and the base curve after a finite étale bas...

In this paper, we study a spacelike (timelike) ruled W-surface in Minkowski 3-space which satisfies nontrivial relation between elements of the set {K, KII , H, HII}, where (K,H) and (KII , HII) are the Gaussian and mean curvatures of the first and second fundamental forms, respectively. Finally, some examples are constructed and plotted.