6
$\begingroup$

Let $\ell_n$ where $n\geq 3$ be the configuration of $n$ lines in a plane, such that $n-1$ of them pass through a single point and the last one does not and it intersects rest of the $n-1$ lines. I'm interested in computing the $K_0$ of algebraic vector bundles on these lines. More generally for every $m$ I'd like to calculate $K_0$ of $\mathbb{A}^m\times \ell_n$. I expect this to be $\mathbb{A}^1$-invariant for $m>n-2$. I'd appreciate any ideas about how to approach such a computation.

There is paper by "BARRY H. DAYTON AND CHARLES A. WEIBEL1". They compute $K$-theory of hyperplanes in general position which satisfy some conditions. None of my cases above satisfy it except $\ell_3$ which gives $K_0(\ell_3)\cong K_0(k)\oplus K_1(k)$ where $k$ is the base field.

Improve this question
$\endgroup$
1
  • 1
    $\begingroup$ For singular curves, the nontrivial part of $K_0(X)$ should come from the Picard group $Pic(X)$ which can be computed in terms of the local structure of singular points. One reference about Picard group of singular curves is link.springer.com/content/pdf/10.1007/BF01440949.pdf. $\endgroup$
    – Evgeny Shinder
    Commented Sep 9, 2020 at 23:10

0

Reset to default

You must log in to answer this question.