Speyer and Sturmfels [SpSt] associated Gröbner toric degenerations Gr2(C) of Gr2(Cn) to each trivalent tree T with n leaves. These degenerations induce toric degenerations M r of Mr, the space of n ordered, weighted (by r) points on the projective line. Our goal in this paper is to give a geometric (Euclidean polygon) description of the toric fibers as stratified symplectic spaces and describe ...

If a finite partial projective plane can be extended to a projective plane of order n, then it can be extended by a sequence of one line extensions. We describe necessary conditions for such an extension to exist, and restrictions which can be imposed on the intermediate sequence of extensions. This method can be used to attempt to construct a projective plane of non-prime-power order. 1. Parti...

We develop an intersection theory for a singular hermitian line bundle with positive curvature current on a smooth projective variety and irreducible curves on the variety. And we prove the existence of a natural rational fibration structure associated with the singular hermitian line bundle. Also for any pseudoeffective line bundle on a smooth projective variety, we prove the existence of a na...

The paper gives a practical rapid algorithm for doing projective reconstruction of a scene consisting of a set of lines seen in three or more images with uncalibrated cameras. The algorithm is evaluated on real and ideal data to determine its performance in the presence of varying degrees of noise. By carefully consideration of sources of error, it is possible to get accurate reconstruction wit...

We will generalize the projective model structure in the category of unbounded complexes of modules over a commutative ring to the category of unbounded complexes of quasi-coherent sheaves over the projective line. Concretely we will define a locally projective model structure in the category of complexes of quasicoherent sheaves on the projective line. In this model structure the cofibrant obj...

We establish a Yau–Tian–Donaldson type correspondence, expressed in terms of single Delzant polytope, concerning the existence extremal Kähler metrics on large class toric fibrations, introduced by Apostolov–Calderbank–Gauduchon–Tonnesen-Friedman and called semi-simple principal fibrations. use that an metric total space corresponds to weighted constant scalar curvature (in sense Lahdili) corre...