Related papers: Monomial resolutions of morphisms of algebraic sur…

Suppose that $\Phi:X\to Y$ is a morphism from a 3 fold to a surface (over an algebraically closed field of characteristic zero). We prove that there exist sequences of blowups of nonsingular subvarieties $X_1\to X$ and $Y_1\to Y$ such that…

Suppose that X to Y is a generically finite map of nonsingular varieties over a field of characteristic zero, and v is a valuation of the function field of X. We prove that it is possible to perform a sequence of monoidal transforms X' to X…

Suppose that $f:X\to Y$ is a dominant morphism of 3-folds over an algebraically closed field of characteristic zero. We prove that there exist sequences of blow ups of points and nonsingular curves $\Phi:X_1\to X$ and $\Psi:Y_1\to Y$ such…

We prove that any dominant morphism of algebraic varieties over a field k of characteristic zero can be transformed into a toroidal (hence monomial) morphism by projective birational modifications of source and target. This was previously…

A monomial algebra is the quotient of a polynomial algebra by an ideal generated by monomials. We prove that finite-dimensional monomial algebras are characterized by their automorphism group among finite-dimensional, local algebras with…

In this note we prove that any affine algebraic monoid can be obtained as the endomorphisms' monoid of a finite-dimensional (nonassociative) algebra.

Suppose that f is a dominant morphism from a k-variety X to a k-variety Y, where k is a field of characteristic 0 and v is a valuation of the function field k(X). We allow v to be an arbitary valuation, so it may not be discrete. We prove…

In this paper, we give a new proof of the foundational result, due to S. Cutkosky, on the existence of a monomialisation of a morphism from a 3-fold to a surface. Our proof brings to the fore the notion of log-Fitting ideals, and requires…

Let $f:X \to Y$ be a proper morphism of normal varieties with $f_*\mathcal{O}_X = \mathcal{O}_Y$. If $X$ is toric, then $Y$ is toric and $f$ is a toric morphism for some toric structures on $X$ and $Y$.

Given an ideal $\mathcal I$ on a variety $X$ with toroidal singularities, we produce a modification $X' \to X$, functorial for toroidal morphisms, making the ideal monomial on a toroidal stack $X'$. We do this by adapting the methods of…

The problem of toroidalization is to construct a toroidal lifting of a dominant morphism $\varphi:X\to Y$ of algebraic varieties by blowing up in the target and domain. This paper contains a solution to this problem when $\varphi$ is…

In this paper we prove strong toroidalization of birational morphisms of 3-folds. Suppose that f:X\to Y is a birational morphism of nonsingular complete 3-folds, and D_Y, D_X are simple normal crossings divisors on Y and X such that…

The toroidalization conjecture of D. Abramovich, K. Karu, K. Matsuki, and J. Wlodarczyk asks whether any given morphism of nonsingular varieties over an algebraically closed field of characteristic zero can be modified into a toroidal…

We prove that for any singular integral affine variety $X$ of finite presentation over a perfect field defined over $\mathbb Z$, there exists a smooth morphism from $Y$ onto $X$ such that $Y$ admits a resolution. That is, there exists a…

Let A^2 be the affine plane over a field K of characteristic 0. Birational morphisms of A^2 are mappings A^2 \to A^2 given by polynomial mappings \phi of the polynomial algebra K[x,y] such that for the quotient fields, one has K(\phi(x),…

A sequence of monoidal transformations is defined, in terms of invariants, for a singular hypersurface embedded in a smooth scheme of positive characteristic. Some examples are added to illustrate the improvement of singularities by this…

We give a simpler and more conceptual proof that a morphism from a 3-fold to a surface, over an algebraically closed field of characteristic 0, can be made into a toroidal morphism by sequences of blow ups of nonsingular subvarieties above…

It is shown that if a finite generically smooth morphism $f\,:\,Y\,\longrightarrow\, X$ of smooth projective varieties induces an isomorphism of the \'etale fundamental groups, then the induced map of the stratified fundamental groups…

Let $K$ be a field of characteristic zero, $X$ and $Y$ be smooth $K$-varieties, and let $G$ be a algebraic $K$-group. Given two algebraic morphisms $\varphi:X\rightarrow G$ and $\psi:Y\rightarrow G$, we define their convolution…

We consider a finite \'etale morphism $f:Y \to X$ of quasi-smooth Berkovich curves over a complete nonarchimedean non-trivially valued field $k$, assumed algebraically closed and of characteristic 0, and a skeleton…