Let $X$ be a smooth, complex projective algebraic variety defined over a number field $K$.

Let $D$ be a divisor of $X$ defined over $K$ with the following property:

For any curve $C$ defined over $K$, we have $operatorname{deg (D_{|C})=0}$

Is it then true that $c_1(D)=0$?

In general, in order to have $c_1(D)=0$, I should check that $operatorname{deg (D_{|C})=0}$

for *any curve* (not just the ones defined over $K$). I’m asking if in this particular setting, the curves defined over $K$ are enough.

MathOverflow Asked by manifold on January 28, 2021

1 AnswersEvery curve on $X$ is algebraically equivalent to a curve defined over a finite extension of $K$, and then a union of Galois conjugates will be defined over $K$. So, if you allow reducible curves, then the answer is yes.

Added: The intersection product is Galois invariant.

For a nonperfect field $k$ and a divisor $D$ defined over a purely inseparable extension of $k$ of degree $p^m$, the divisor $p^m D$ is defined over $k$.

Regard $D$ as the Cartier divisor defined by a family of pairs $(f_{i},U_{i}^{prime})$, $f_{i}in k^{prime}(X)$, and let $U_{i}$ be the image of $U_{i}^{prime}$ in $X$; then $k^{prime}(X)^{p^{m}}subset k(X)$, and so the pairs $(f_{i}% ^{p^{m}},U_{i})$ define a divisor on $X$ whose inverse image on $X_{k^{prime}}$ is $p^{m}D$.

Correct answer by anon on January 28, 2021

0 Asked on December 15, 2021 by inkspot

ag algebraic geometry reference request rt representation theory

1 Asked on December 15, 2021 by chris-ramsey

fa functional analysis linear algebra matrices oa operator algebras

5 Asked on December 13, 2021 by william-stagner

ct category theory ho history overview kt k theory and homology rt representation theory symmetric functions

1 Asked on December 13, 2021

1 Asked on December 13, 2021

2 Asked on December 13, 2021 by mirco-a-mannucci

0 Asked on December 13, 2021

1 Asked on December 13, 2021 by jdoe2

3 Asked on December 13, 2021 by ikp

0 Asked on December 13, 2021

1 Asked on December 13, 2021 by andi-bauer

ct category theory monoidal categories symmetric monoidal categories

1 Asked on December 13, 2021 by user45397

0 Asked on December 13, 2021

ca classical analysis and odes differential equations fa functional analysis real analysis rough paths

0 Asked on December 13, 2021

dg differential geometry geometric measure theory measure theory numerical analysis of pde oc optimization and control

2 Asked on December 13, 2021

1 Asked on December 11, 2021

fa functional analysis nuclear spaces short exact sequences tensor products

1 Asked on December 11, 2021

dg differential geometry lie groups random matrices rt representation theory st statistics

0 Asked on December 11, 2021

0 Asked on December 11, 2021 by titouan-vayer

1 Asked on December 11, 2021

foundations lo logic mathematical philosophy proof theory reference request

Get help from others!

Recent Questions

- MouseLook Script “Pops” back to the last value when the script is enabled after being disabled or destroyed
- Unity app crashes when using unmodified custom Android manifest (didn’t find class “UnityPlayerActivity”)
- How do i draw a ray in unity
- How to test consistency of responses?
- How can I understand these variograms?

Recent Answers

- Philipp on How do i draw a ray in unity
- kjetil b halvorsen on How to test consistency of responses?
- eric_kernfeld on How to test consistency of responses?
- DMGregory on MouseLook Script “Pops” back to the last value when the script is enabled after being disabled or destroyed
- Justin Markwell on Unity app crashes when using unmodified custom Android manifest (didn’t find class “UnityPlayerActivity”)

© 2022 AnswerBun.com. All rights reserved.