We introduce a new concept in the area of formal logic: axioms for model morphisms. We work in the setting of specification languages that define the semantics of a theory as a category of models. While it is routine to use axioms to specify the class of models of a theory, there has so far been no analogue to systematically specify the morphisms between these models. This leads to subtle probl...

We prove that the intersection cohomology (together with perverse and Hodge filtrations) for moduli space of one-dimensional semistable sheaves supported in an ample curve class on a toric del Pezzo surface is independent Euler characteristic sheaves. also analogous result Higgs bundles respect to effective divisor $D$ degree $\mathrm{deg}(D)>2g-2$. Our results confirm cohomological $\chi$-inde...

1.1. Crash course in blowing up. Last time I began to talk about blowing up. Let X be a scheme, and I ⊂ OX a sheaf of ideals on X. (Technical requirement automatically satisfied in our situation: I should be a coherent sheaf, i.e. finitely generated.) Here is the “universal property” definition of blowing-up. Then the blow-up of OX along I is a morphism π : X̃ → X satisfying the following univer...

We describe and prove uniqueness of a natural homomorphism (called the Orchard morphism) between some groups associated to finite sets.

208 NOTICES OF THE AMS VOLUME 51, NUMBER 2 As far as I know, the notion of a quasi-morphism does not have much to do with category theory. This very natural idea underlies several interesting recent developments at the crossroads of algebra, topology, geometry, and dynamics. Other, perhaps better, names currently in use for the same concept are quasi-homomorphism and pseudo-character. Let G be ...