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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 …

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 …

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 …

K-equivalence $\implies$ isomorphism of (rational) Chow motives is a conjecture going back to this 2002 paper of Wang (Conjecture 2.2); see this overview paper of his for what else K-equivalence shoul …

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 …

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 …

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 …

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 …

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 …

Nikulin, in $\S$2 of his classical paper about lattices explains an algorithm how to compute the Milnor lattice of a function germ with an isolated singularity. Specifically, in Theorem 2.2.2 he says …

Welcome to MathOverflow! Question 1: this can be checked locally, on affine opens or local rings, and then becomes an exercise: if $0 \to M' \to M \overset{\pi}{\to} M'' \to 0$ is an exact sequence of …

There are no simple examples as yet; it's been an open question going back to at least Larsen and Lunts - Motivic measures and stable birational geometry, which has been open for about 15 years, and s …

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 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 …

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 …