Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

An example for Kuznetsov-Shinder conjecture

Tanya Kaushal Srivastava

TL;DR

The paper addresses how derived equivalence between smooth projective varieties interacts with motivic invariants in the Grothendieck ring of varieties and the Lefschetz motive, in particular whether $D$-equivalence forces $L$-equivalence or stronger. It constructs an explicit infinite family of simply connected rational 3-folds $X_{ extbf{p}}$ by blowing up eight points in $ abla P^3$ such that $D(X_{ extbf{p}})rom D(X_{ extbf{q}})$ for $ extbf{p} eq extbf{q}$ but $X_{ extbf{p}} otrom X_{ extbf{q}}$. Remarkably, all members share the same class in $K_0(Var/k)$, hence are $lam$-equivalent, which provides evidence for the Kuznetsov–Shinder conjecture in the simply connected, rational case. The discussion also connects this to birational invariants via Larsen–Lunts and to Kawamata's DK-conjecture, showing the constructed family satisfies these links via a common blow-up and canonical-divisor pull-backs. The significance is that it yields an explicit, infinite family of derived-equivalent but non-isomorphic varieties with identical motivic classes, clarifying the interplay between derived categories, motivic invariants, and birational geometry.

Abstract

We give an example for the Kuznetsov-Shinder conjecture, with infinitely many non-isomorphic D-equivalent and L-equivalent varieties.

An example for Kuznetsov-Shinder conjecture

TL;DR

The paper addresses how derived equivalence between smooth projective varieties interacts with motivic invariants in the Grothendieck ring of varieties and the Lefschetz motive, in particular whether -equivalence forces -equivalence or stronger. It constructs an explicit infinite family of simply connected rational 3-folds by blowing up eight points in such that for but . Remarkably, all members share the same class in , hence are -equivalent, which provides evidence for the Kuznetsov–Shinder conjecture in the simply connected, rational case. The discussion also connects this to birational invariants via Larsen–Lunts and to Kawamata's DK-conjecture, showing the constructed family satisfies these links via a common blow-up and canonical-divisor pull-backs. The significance is that it yields an explicit, infinite family of derived-equivalent but non-isomorphic varieties with identical motivic classes, clarifying the interplay between derived categories, motivic invariants, and birational geometry.

Abstract

We give an example for the Kuznetsov-Shinder conjecture, with infinitely many non-isomorphic D-equivalent and L-equivalent varieties.
Paper Structure (4 sections, 2 theorems, 5 equations)

This paper contains 4 sections, 2 theorems, 5 equations.

Table of Contents

  1. The Grothendieck ring of varieties
  2. An example in support: Rational 3-folds
  3. Grothendieck Ring of Varieties and Birationality
  4. Comparison with K-equivalence

Key Result

Theorem 1.6

Lesi Let $\textbf{p}$ denote an ordered $8$-tuple of distinct points in $\mathbb P^3$, and let $X_{\textbf{p}}$ be the blow-up of $\mathbb P^3$ at the points of $\textbf{p}$. There is an infinite set $W$ of configurations of $8$ points in $\mathbb P^3$ such that if $\textbf{p}$ and $\textbf{q}$ are

Theorems & Definitions (3)

  • Theorem 1.6
  • proof
  • Theorem 1.12