Related papers: Notes on Nash modification

The subject is partial resolution of singularities. Given an algebraic variety X (not necessarily equidimensional) in characteristic zero (or, more generally, a pair (X,D), where D is a divisor on X), we construct a functorial…

The Nash multiplicity sequence was defined by M. Lejeune-Jalabert as a non-increasing sequence of integers attached to a germ of a curve inside a germ of a hypersurface. M. Hickel generalized this notion and described a sequence of blow ups…

When $X$ is a $d$-dimensional variety defined over a field $k$ of characteristic zero, a constructive resolution of singularities can be achieved by successively lowering the maximum multiplicity via blow ups at smooth equimultiple centers.…

Configuration polynomials generalize the Kirchhoff polynomial of a graph, as well as the Symanzik polynomials that appear in the denominators of Feynman integrands. The configuration hypersurfaces cut out by such polynomials are typically…

Given a projective contraction $\pi \colon X\rightarrow Z$ and a log canonical pair $(X, B)$ such that $-(K_X+B)$ is nef over a neighborhood of a closed point $z\in Z$, one can define an invariant, the complexity of $(X, B)$ over $z \in Z$,…

Given a variety $X$ over a perfect field, we study the partition defined on $X$ by the multiplicity (into equimultiple points), and the effect of blowing up at smooth equimultiple centers. Over fields of characteristic zero we prove…

Let $X$ be a normal complex variety and $\pi:\tilde X \to X$ a resolution of singularities. We show that the inclusion morphism $\pi_*\Omega_{\tilde X}^p\hookrightarrow \Omega_X^{[p]}$ is an isomorphism for $p <…

Let $X$ be a closed subscheme embedded in a scheme $W$ smooth over a field ${\bf k}$ of characteristic zero, and let ${\mathcal I}(X)$ be the sheaf of ideals defining $X$. Assume that the set of regular points of $X$ is dense in $X$. We…

Suppose $X$ is an irreducible complex variety. We show that when $X$ is ruled, the group of birational transformations $Bir(X)$, as a group, determines $X$ up to birational transformations and automorphisms of the base field. In contrast,…

Let $X$ be a singular algebraic variety defined over a field $k$, with quotient field $K(X)$. Let $s \geq 2$ be the highest multiplicity of $X$ and $F_s(X)$ the set of points of multiplicity $s$. If $Y\subset F_s(X)$ is a regular center and…

We consider the problem of finding an ideal whose blowup defines the Nash blowup of a toric surface and such that its zero locus coincides with the singular set of the toric surface.

We study different notions of blow-up of a scheme X along a subscheme Y, depending on the datum of an embedding of X into an ambient scheme. The two extremes in this theory are the ordinary blow-up, corresponding to the identity, and the…

We strengthen Gabber's $l'$-alteration theorem by avoiding all primes invertible on a scheme. In particular, we prove that any scheme $X$ of finite type over a quasi-excellent threefold can be desingularized by a…

In this paper we give a criterion for an isolated, hypersurface singularity of dimension $n\ (\geq 2)$ to have the canonical modification by means of a suitable weighted blow-up. Then we give a counter example to the following conjecture by…

Let $H$ and $H'$ be two ample line bundles over a smooth projective surface $X$, and $M(H)$ (resp. $M(H')$) the coarse moduli scheme of $H$-semistable (resp. $H'$-semistable) sheaves of fixed type $(r,c_1,c_2)$. We construct a sequence of…

We will describe a one-step "Gorensteinization" process for a Schubert variety by blowing-up along its boundary divisor. The local question involves Kazhdan-Lusztig varieties which can be degenerated to affine toric schemes defined using…

In this paper we characterize the Blowing-up maps of ordinary singularities for which there exists a natural Gysin morphism, i.e. a bivariant class $\theta \in Hom_{D(Y)}(R\pi_*\mathbb Q_X, \mathbb Q_Y)$, compatible with pullback and with…

We study the higher Nash blow-ups introduced by T. Yasuda and investigate the higher version of the classical Nobile's theorem. In particular, we give a characteristic free proof of the higher Nobile's theorem for the graded case. We also…

We classify smooth toric Fano varieties of dimension $n\geq 3$ containing a toric divisor isomorphic to $\PP^{n-1}$. As a consequence of this classification, we show that any smooth complete toric variety $X$ of dimension $n\geq 3$ with a…

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…