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Let $G$ be a finite group acting linearly on $\mathbb{A}^n$. Do we expect algebraic vector bundles on $X := \mathbb{A}^n/G$ to be trivial? Here by the quotient I mean the singular scheme, not the stac …

An old conjecture of Bondal–Orlov–Kawamata predicts that K-equivalent varieties are D-equivalent, see Kawamata's paper D-equivalence and K-equivalence for definitions. In particular this applies to bi …

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 …

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 …

Background. The threefold compound du Val singularities have been introduced by Miles Reid in the 1980s [R1, R2, R3]. Their geometric description is that a general hyperplane section through the singu …

General elliptic K3 surfaces. Consider K3 surfaces of Picard rank two with Neron-Severi lattice isomorphic to $$\left[\begin{array}{cc} 2d & t \\ t & 0 \end{array}\right]$$ for some positive integers …

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 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 …

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 …

I am looking for smooth projective varieties $X$, with $h^i(X, \mathcal{O}_X) = 0$ for $i > 0$, with a very ample line bundle $L$ with some nonvanishing higher cohomology. What is clear: (1) Curves wi …

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 …

Here is an expanded version of my comments. Let's work over the complex numbers which I suppose is assumed in the question. Let $K_0(Var)$ be the Grothendieck ring of varieties (see e.g. https://arxiv …

One can construct t-structures on the bounded derived category of coherent sheaves on a smooth projective curve (or higher-dimensional variety) by tilting, see Bayer's notes, Prop. 3.6.1, and the corr …

In characteristic zero $[\mathrm{Spec}(K)] = [\mathrm{Spec}(K')]$ for finite field extensions of $k$ implies that $K$ and $K'$ are isomorphic. Indeed, by the Larsen-Lunts theorem for smooth projectiv …

I am expanding naf's comments to make a self-contained community wiki answer. By an elliptic fibration we mean a smooth projective relatively minimal surface $f: X \to C$ with general fiber given by a …