a toroidalization of a dominant morphism $varphi: xto y$ of algebraic varieties over a field of characteristic zero is a toroidal lifting of $varphi$ obtained by performing sequences of blow ups of nonsingular subvarieties above $x$ and $y$. we give a proof of toroidalization of locally toroidal morphisms of 3-folds.

The geometry of spaces of rational curves of low degree on Fano threefolds is a very active area of algebraic geometry. One of the main goals is to understand the Abel-Jacobi morphism from the base of a space of such curves to the intermediate Jacobian of the threefold. For instance, does a given space of curves dominate the intermediate Jacobian? If so, what are the fibers of this morphism? Do...

It is known that each word of length n contains at most + 1 distinct palindromes. A finite rich a with maximal number palindromic factors. The definition richness can be naturally extended to infinite words. Sturmian words and Rote complementary symmetric sequences form two classes binary words, while episturmian coding d-interval exchange transformations give us other examples on larger alphab...

This paper is devoted to the study of algebraic properties of rings of continuous functions. Our aim is to show that these rings, even if they are highly nonnoetherian, have properties quite similar to the elementary properties of noetherian rings: we give going-up and going-down theorems, a characterization of z-ideals and of primary ideals having as radical a maximal ideal and a flatness crit...

In this paper C denotes a category and o1, o2, o3 denote objects of C. Let C be a non empty category structure with units and let o be an object of C. Observe that 〈o, o〉 is non empty. The following propositions are true: (1) Let v be a morphism from o1 to o2, u be a morphism from o1 to o3, and f be a morphism from o2 to o3. If u = f · v and f −1 · f = id(o2) and 〈o1, o2〉 6= ∅ and 〈o2, o3〉 6= ∅...

Abstract. We introduce the concept of a morphism between coloured nets. Our definition generalizes Petris definition for ordinary nets. A morphism of coloured nets maps the topological space of the underlying undirected net as well as the kernel and cokernel of the incidence map. The kernel are flows along the transitionbordered fibres of the morphism, the cokernel are classes of markings of th...

0.1. The problem. Roughly speaking, the semistable reduction problem we address here asks for the following: Let X → B be a surjective morphism of complex projective varieties with geometrically integral generic fiber. Find a generically finite proper surjective morphism (that is, an alteration) B1 → B, and a proper birational morphism (that is, a modification) Y → X ×B B1, such that the morphi...

We continue our study on infinitesimal lifting properties of maps between locally noetherian formal schemes started in [3]. In this paper, we focus on some properties which arise specifically in the formal context. In this vein, we make a detailed study of the relationship between the infinitesimal lifting properties of a morphism of formal schemes and those of the corresponding maps of usual s...