3-folds in \({\mathbb{P}}^ 5\) of degree 12
The classification of (smooth) codimension-two subvarieties in \(\mathbb{P}^ 4\) and \(\mathbb{P}^ 5\) has gained interest since Ellingsrud and Peskine proved the finiteness of the number of components of the Hilbert scheme of non-general-type surfaces in \(\mathbb{P}^ 4\) (the analogous statement for \(\mathbb{P}^ 5\) has been proven by Braun, Ottaviani, Schneider and Schreyer). Two present research lines (too far from being convergent for the time being) are the classification of codimension-two subvarieties in \(\mathbb{P}^ 4\) and \(\mathbb{P}^ 5\) of small degree and, on the other hand, the search of an effective degree bound for non-general-type codimension-two subvarieties in \(\mathbb{P}^ 4\) and \(\mathbb{P}^ 5\). In the paper under review, the author gives one more step in the classification of smooth 3-folds in \(\mathbb{P}^ 5\), by treating the degree 12 case. More precisely, he gives a list of eight possible families and proves the actual existence of 3-folds for seven of the cases, the last case remaining undecided. In order to determine the invariants of the possible 3-folds, adjunction theory methods are used, and many possibilities need to be ruled out.
number of components of the Hilbert scheme of non-general-type surfaces
scientific article; zbMATH DE number 572235
scientific article