We prove that if S is a smooth reflexive surface in P 3 defined over a finite field F q , then there exists an F q -line meeting S transversely provided that q ≥ c deg(S), where c = 3+ √ 17 4 ≈ 1.7808. Without the reflexivity hypothesis, we prove the existence of a transverse F q -line for q ≥ deg(S) 2 . √ 17 4 ≈ 1.780776. Recall that a line L meets a surface S transversely if the intersection S ∩ L consists of d = deg(S) distinct geometric points. The reflexivity

We study the Galois action attached to the Dwrok surfaces X λ : X 4 0 + X 4 1 + X 4 2 + X 4 3 − 4λX 0 X 1 X 2 X 3 = 0 with parameter λ in a number field F . We show that when X λ has geometric Picard number 19, its Néron-Severi group N S(X λ ) ⊗ Q is a direct sum of quadratic characters. We provide two proofs to this conclusion in our article. In particular, the geometrically proof determines the conductor of each of quadratic characters. Our result matches the one in [DKS + 18]. With this decomposition, we give another proof to a result of Wan [Wan06].

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Let K/k be a finite Galois extension of number fields, and let H K be the Hilbert class field of K. We find a way to verify the nonsplitting of the short exact sequenceby finite calculation. Our method is based on the study of the principal version of the Chebotarev density theorem, which represents the density of the prime ideals of k that factor into the product of principal prime ideals in K. We also find explicit equations to express the principal density in terms of the invariants of K/k. In particular, we prove that the group structure of the ideal class group of K can be determined by reading the principal densities.

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