TY - JOUR
T1 - The Noether–Lefschetz theorem in arbitrary characteristic
AU - Ji, Lena
N1 - the author was supported by the National Science Foundation Graduate Research Fellowship Program under Grant No. DGE-1656466. The author is grateful to her advisor, J\u00E1nos Koll\u00E1r, for his wisdom, patience, and generosity. She would also like to thank Claire Voisin for granting permission to use her proof of Proposition 4.6; Robin Hartshorne for helpful comments and questions, especially regarding Section 4.1; Bjorn Poonen for suggesting the extension to other fields in Section 4.3; Joe Waldron for answering many questions and for comments on an early draft of this article; Scott Nollet for suggestions on the presentation; Remy van Dobben de Bruyn and Takumi Murayama for helpful conversations; and the referee for their helpful suggestions.
PY - 2024
Y1 - 2024
N2 - We show that if X ⊂ PNk is a normal variety of dimension ≥ 3 and H ⊂ PNk a very general hypersurface of degree d = 4 or ≥ 6, then the restriction map Cl(X) → Cl(X ∩ H) is an isomorphism up to torsion. If dim X ≥ 4, the result holds for d ≥ 2. The proof uses the relative Jacobian of a curve fibration, together with a specialization argument, and the result holds over fields of arbitrary characteristic.
AB - We show that if X ⊂ PNk is a normal variety of dimension ≥ 3 and H ⊂ PNk a very general hypersurface of degree d = 4 or ≥ 6, then the restriction map Cl(X) → Cl(X ∩ H) is an isomorphism up to torsion. If dim X ≥ 4, the result holds for d ≥ 2. The proof uses the relative Jacobian of a curve fibration, together with a specialization argument, and the result holds over fields of arbitrary characteristic.
UR - https://www.scopus.com/pages/publications/85189703737
UR - https://www.scopus.com/pages/publications/85189703737#tab=citedBy
U2 - 10.1090/jag/825
DO - 10.1090/jag/825
M3 - Article
AN - SCOPUS:85189703737
SN - 1056-3911
VL - 33
SP - 567
EP - 600
JO - Journal of Algebraic Geometry
JF - Journal of Algebraic Geometry
IS - 3
ER -