On the lifting problem in positive characteristic (Q2809911)

Let \(X\subset \mathbb P^n\) be an integral projective variety of codimension \(2\) and \(Y=X\cap H\) be a general hyperplane section of \(X\). The lifting problem asks for conditions ensuring that a hypersurface \(C\) in \(H\) containing \(Y\) can be lifted to a hypersurface of the same degree in \(\mathbb P^n\) containing \(X\). The celebrated generalized trisecant lemma of Laudal, successively extended by Gruson-Peskine, solves the case \(n=3\) in characteristic \(0\) (see [\textit{O. A. Laudal}, Lect. Notes Math. 687, 112--149 (1978; Zbl 0392.14002)], [\textit{L. Gruson} and \textit{C. Peskine}, Prog. Math. 24, 33--35 (1982; Zbl 0526.14021)]. Also if \(n= 4,5,6\) the problem is solved in characteristic \(0\), thanks to results of many authors including \textit{E. Mezzetti} and \textit{I. Raspanti} [Rend. Semin. Mat., Torino 48, No. 4, 529--537 (1990; Zbl 0779.14010)], \textit{E. Mezzetti} [J. Algebr. Geom. 3, No. 3, 375--398 (1994; Zbl 0833.14005)] and \textit{M. Roggero} [Lect. Notes Pure Appl. Math. 217, 309--326 (2001; Zbl 1065.14064)].NEWLINENEWLINEIn positive characteristic, the author of this article has obtained analogous results for \(n=3,4\) (see [J. Algebr. Geom. 18, No. 3, 459--475 (2009; Zbl 1168.14026); Int. Electron. J. Geom. 5, No. 1, 128--139 (2012; Zbl 1308.14035)]). Here she surveys her results, explaining the methods used in the proofs, and states some open problems in higher dimension.