Grothendieck-Lefschetz type theorems for the local Picard group (Q2871002)

Noether-Lefschetz theory studies the line bundles \(L\) on spaces \(X\) for which the restriction map \(\text{Pic} X \to \text{Pic} Y\) is an isomorphism for general \(Y \in |H^0(X,L)|\). This theory has seen many developments since Lefschetz proved Noether's theorem in the 1920s (see survey of \textit{J. Brevik} and \textit{S. Nollet} [Contemp. Math. 608, 21--50 (2014; Zbl 1314.14016)]). The paper under review deals with the local version: the local Picard group of \(X\) at \(x \in X\) is \(\text{Pic}^{\text{loc}}_x X = \text{Pic} (\text{Spec} {\mathcal O}_{x,X} \setminus \{x\})\). For \(X\) normal and \(x \in X\) a closed point contained in a Cartier divisor \(Y \subset X\) with \(\dim_x X \geq 4\), the author considers the kernel \(K\) of the restriction map NEWLINE\[NEWLINE r: \text{Pic}^{\text{loc}}_x X \to \text{Pic}^{\text{loc}}_x Y, NEWLINE\]NEWLINE conjecturing that (A) \(K\) is contained in the connected subgroup of \(\text{Pic}^{\text{loc}}_x X\), (B) \(K=0\) if \(Y\) satisfies Serre's condition \(S_2\) and (C) \(K\) is \(p\)-torsion if \(X\) is an excellent local \(\mathbb F_p\)-algebra. Conjecture (B) would strengthen Grothendieck's result that \(K =0\) if \(\text{depth}_x {\mathcal O}_X \geq 4\) [\textit{A. Grothendieck}, Séminaire de géométrie algébrique par Alexander Grothendieck 1962. Cohomologie locale des faisceaux cohérents et théorèmes de Lefschetz locaux et globaux. Fasc. I. Exposés I à VIII; Fasc. II. Exposés IX à XIII. 3ieme édition, corrigée. Bures-Sur-Yvette (Essonne): Institut des Hautes Études Scientifiques (1962; Zbl 0159.50402)]. The author poses interesting problems and approaches whose solutions would prove conjecture (B) over \(\mathbb C\).NEWLINENEWLINEThe author proves Conjecture (A) under conditions sufficient to study higher dimensional analogs of phenomena for deformation theory of \(2\)-dimensional surface singularities. Specifically, (1) Lee and Park produced projective surfaces \(S\) with quotient singularities and an ample canonical class which smooth to rational surfaces [\textit{Y. Lee} and \textit{J. Park}, Invent. Math. 170, No. 3, 483--505 (2007; Zbl 1126.14049)], (2) Mumford produced non-normal isolated smoothable surface singularities whose normalization is simple elliptic [\textit{D. Mumford}, in: C.P. Ramanujam. - A tribute. Collect. Publ. of C.P. Ramanujam and Pap. in his Mem., Tata Inst. fundam. Res., Stud. Math. 8, 247--262 (1978; Zbl 0444.14002)] and (3) Every rational surface singularity has a smoothing that admits a simultaneous resolution by work of [\textit{M. Artin}, J. Algebra 29, 330--348 (1974; Zbl 0292.14013)]: these form the Artin component of the deformation space. The author shows that none of these two dimensional results hold for isolated singularities in higher dimension.