DOI: 10.1017/nmj.2026.10139 ISSN: 0027-7630

Noether–Lefschetz theory and surfaces containing a multiple curve

John Brevik, Scott Nollet
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Abstract

We show that if X is a regular complex projective threefold and Y is a very general surface of sufficiently high degree containing a thick infinitesimal neighborhood Z of a smooth curve C , then the restriction map

r : Pic   X → Pic   Y $r: \textrm{Pic}\ X \to \textrm{Pic}\ Y$ r colon Pic upper X right arrow Pic upper Y
is an isomorphism. Our proof uses Moishezon’s Noether–Lefschetz theorem and some general tools of Hartshorne and Polini. We give the complete picture for all d when
X = P 3 $X = {\mathbb P}^3$ upper X equals double struck upper P cubed
, C is a line, and
m = 2 $m=2$ m equals 2
, using a classification of Urabe to handle the difficult case of quartic surfaces.

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