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Vik78
Vik78
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What are some examples of smooth projective varieties $X$ over a finite field for which the Tate conjecture for divisors is known, and which admit a smooth morphism to a smooth projective curve (equivalently to $\mathbb{P}^1$)? I am not really interested in the case where $X$ is an abelian variety and the map to the curve factors through an elliptic curve. Unfortunately I do not think K3 surfaces can be smooth over a curve.

EDIT: removed redundant flatness assumption and incorrect assumption that it reduces to the case where the base is $\mathbb{P}^1$

What are some examples of smooth projective varieties $X$ over a finite field for which the Tate conjecture for divisors is known, and which admit a smooth morphism to a smooth projective curve (equivalently to $\mathbb{P}^1$)? I am not really interested in the case where $X$ is an abelian variety and the map to the curve factors through an elliptic curve. Unfortunately I do not think K3 surfaces can be smooth over a curve.

EDIT: removed redundant flatness assumption

What are some examples of smooth projective varieties $X$ over a finite field for which the Tate conjecture for divisors is known, and which admit a smooth morphism to a smooth projective curve? I am not really interested in the case where $X$ is an abelian variety and the map to the curve factors through an elliptic curve. Unfortunately I do not think K3 surfaces can be smooth over a curve.

EDIT: removed redundant flatness assumption and incorrect assumption that it reduces to the case where the base is $\mathbb{P}^1$

Edited to remove redundant flatness assumption
Source Link
Vik78
Vik78
  • 658
  • 4
  • 11

What are some examples of smooth projective varieties $X$ over a finite field for which the Tate conjecture for divisors is known, and which admit a smooth flat morphism to a smooth projective curve (equivalently to $\mathbb{P}^1$)? I am not really interested in the case where $X$ is an abelian variety and the map to the curve factors through an elliptic curve. Unfortunately I do not think K3 surfaces can be smooth and flat over a curve.

EDIT: removed redundant flatness assumption

What are some examples of smooth projective varieties $X$ over a finite field for which the Tate conjecture for divisors is known, and which admit a smooth flat morphism to a smooth projective curve (equivalently to $\mathbb{P}^1$)? I am not really interested in the case where $X$ is an abelian variety and the map to the curve factors through an elliptic curve. Unfortunately I do not think K3 surfaces can be smooth and flat over a curve.

What are some examples of smooth projective varieties $X$ over a finite field for which the Tate conjecture for divisors is known, and which admit a smooth morphism to a smooth projective curve (equivalently to $\mathbb{P}^1$)? I am not really interested in the case where $X$ is an abelian variety and the map to the curve factors through an elliptic curve. Unfortunately I do not think K3 surfaces can be smooth over a curve.

EDIT: removed redundant flatness assumption

Source Link
Vik78
Vik78
  • 658
  • 4
  • 11

Known cases of Tate conjecture for varieties which are smooth over a curve

What are some examples of smooth projective varieties $X$ over a finite field for which the Tate conjecture for divisors is known, and which admit a smooth flat morphism to a smooth projective curve (equivalently to $\mathbb{P}^1$)? I am not really interested in the case where $X$ is an abelian variety and the map to the curve factors through an elliptic curve. Unfortunately I do not think K3 surfaces can be smooth and flat over a curve.