# First Chern class and field extensions

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

Every 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

## Related Questions

### Is there a source in which Demazure’s function $p$ defined in SGA3, exp. XXI, is calculated?

0  Asked on December 15, 2021 by inkspot

### Potential p-norm on tuples of operators

1  Asked on December 15, 2021 by chris-ramsey

### Understanding a quip from Gian-Carlo Rota

5  Asked on December 13, 2021 by william-stagner

### What is Chemlambda? In which ways could it be interesting for a mathematician?

1  Asked on December 13, 2021

### Degree inequality of a polynomial map distinguishing hyperplanes

1  Asked on December 13, 2021

### The inconsistency of Graham Arithmetics plus $forall n, n < g_{64}$

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

### Subgraph induced by negative cycles detected by Bellman-Ford algorithm

0  Asked on December 13, 2021

### How to obtain matrix from summation inverse equation

1  Asked on December 13, 2021 by jdoe2

### Is $arcsin(1/4) / pi$ irrational?

3  Asked on December 13, 2021 by ikp

### A mutliplication for distributive lattices via rowmotion

0  Asked on December 13, 2021

### Tensor product of unit and co-unit in a closed compact category

1  Asked on December 13, 2021 by andi-bauer

### Finite subgroups of $SL_2(mathbb{C})$ arising as a semi-direct product

1  Asked on December 13, 2021 by user45397

### Uniqueness of solutions of Young differential equations

0  Asked on December 13, 2021

### Differentiation under the integral sign for a $L^1$-valued function (shape derivative)

0  Asked on December 13, 2021

### Digraphs with exactly one Eulerian tour

2  Asked on December 13, 2021

### Exactness of completed tensor product of nuclear spaces

1  Asked on December 11, 2021

### An integral with respect to the Haar measure on a unitary group

1  Asked on December 11, 2021

### Best known upper bound for Dedekind zeta function on line $sigma=1$ in the $t$ aspect

0  Asked on December 11, 2021

### Regularity with respect to the Lebesgue measure through dimensions

0  Asked on December 11, 2021 by titouan-vayer

### Reference request on Gentzen’s proof of the consistency of PA

1  Asked on December 11, 2021