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It is known that three-dimensional ordinary double points, that is singular points which complete locally have the equation $xy - zw = 0$ are resolved by a single blow up, with exceptional divisor bei …

For $f(z,w) \in \mathbb{C}[z,w]$ a square-free polynomial, consider an affine threefold $$ X = V(xy + f(z,w)) \subset \mathbb{A}^4. $$ By computing derivatives one sees that the singularities of $X$ a …

This is not known. Motivic integration provides equality of classes of K-equivalent varieties (in particular, for birational with trivial canonical class) in the appropriate localization of the Grothe …

Let $X$ be a complete intersection of a quadric and a cubic in $\mathbb{P}^5$. In the smooth case it is a so-called Fano threefold of index one and degree six. I would like to consider the case when …

Another way to see that the fundamental group is $\mathbb{Z}/k$ is using geometry of cyclic quotient singularities and it goes as follows. We consider the negative twists first. In this case the tot …

Let me try the following argument. First, as you explain one can reduce the general case to the case when $X$ is a curve $C$, possibly singular and reducible. This is because the preimage of $g(\mat …

Here is an argument showing that if $V$ is a smooth complex surface with trivial integral homology groups (note that exotic $\mathbf{A}^2$ do not exist, as explained in the comments), then $[V] = \mat …

In our paper with N. Pavic we have proved that over an algebraically closed field of char. 0, if the weights $a_0, \dots, a_n$ are coprime, so that singularities of $\mathbb{P}(a_0, \dots, a_n)$ are …

Let me summarize what is known about Chern classes and the Chern character on singular varieties, expanding on the comments to the question. On a normal variety the first Chern class is easily defin …

Welcome to Mathoverflow! One can not define intersection product on the Chow groups for a singular variety, even when it is embedded as a divisor in a smooth one: see the quadric cone example in Harts …

The quotients $\mathbf{C}^n / G$ with $G$ finite abelian group (acting linearly) are toric varieties. I present the toric description of the resolution and the discrepancies. If one needs to, one coul …

A toric Cartier divisor $D$ is given by the Cartier data $\{m_\sigma\}_{\sigma \in \Sigma}$ [CLS, Theorem 4.2.8] where for each affine open chart $U_\sigma$, the toric coordinate $x^{-m_\sigma}$ is th …

My question is this: what is the topological analog of the Grothendieck group of coherent sheaves $G(X)$? Background: In Algebra/Algebraic Geometry there are two versions of the Grothendieck group o …

Let $X$ be a smooth genus one curve over $k$. I don't call it elliptic curve because it will have no rational points. By index of $X$ we mean the smallest degree of a closed point on $X$; equivalently …

To clarify what's happening, let us introduce the etale Grothendieck ring varieties $K^{et}(Var/k)$ by imposing the scissor congruence relation AND the relation $[X] = [F][Y]$ for every finite etale c …