All Questions
7 questions
17
votes
2
answers
2k
views
What does taking the graded algebra do to the Grothendieck group, and its relation to the Chow ring?
Let $X$ be a nonsingular variety. (Perhaps some/all of this works over more general smooth schemes, but let's stick to the simple case.)
In, e.g., Fulton's Intersection Theory chapter 15, and Soule's ...
- 156k
4
votes
0
answers
226
views
K-theoretic derivation of Bézout theorem
In the paper "$K$-Theory and Intersection Theory" by Henri Gillet in Handbook of K-theory, 2005 (link behind paywall at Springerlink), the author says:
"When the ground field $k = \mathbb C$, Bézout’...
- 63.9k
9
votes
1
answer
475
views
About Riemann-Roch without denominators
The Riemann-Roch without denominators can be expressed as follows:
Let $f: X\rightarrow Y$ be a closed embedding of quasi-projective smooth $k$-varieties of codimension $d$ for some field $k$. Let $E$ ...
- 2,871
15
votes
1
answer
995
views
Link: Serre's intersection formula <-> Bloch-Quillen Thm / When only intersecting divisors, is there 'shorter' approach of proof known?
In very short:
When proving the equivalence of intersection theory constructed through (Milnor) K-sheaves and their product vs. defining the product via Serre's local multiplicity formula + moving, I ...
- 12.9k
13
votes
1
answer
563
views
Intersection of subvarieties versus ranks of Chow groups modulo numerical equivalences
A nice property of $\mathbb P^n$ is:
Property 1: Two subvarieties $U,V$ such that $\operatorname{dim} U +\operatorname{dim} V \geq n$ always intersect.
(for example, any 2 curves in $\mathbb P^2$ ...
- 7,338
5
votes
0
answers
479
views
where to learn K-group of coherent sheaves modulo numerical equivalence?
I am trying to emerge from my complete ignorance about intersection theory.
I have a bias toward sheaves, so I like the idea of doing intersection theory with the K-group of coherent sheaves. From ...
- 1,889
3
votes
1
answer
581
views
On finite endomorphisms of $\mathbf{P}^r$
This question is basically on applying the Grothendieck-Riemann-Roch theorem to finding a formula for the push-forward of a line bundle on $\mathbf{P}^r$ under a certain morphism. Since I have a lot ...
- 9,497