On reducible hyperplane sections of 4-folds
LANTERI, Antonio ; L. TIRONI, Andrea
J. Math. Soc. Japan, Tome 53 (2001) no. 3, p. 559-563 / Harvested from Project Euclid
We describe 4-dimensional complex projective manifolds $X$ admitting a simple normal crossing divisor of the form $A+B$ among their hyperplane sections, both components $A$ and $B$ having sectional genus zero. Let $L$ be the hyperplane bundle. Up to exchanging the two components, $(X,L,A,B)$ is one of the following: 1) $(X,L)$ is a scroll over $P^{1}$ with $A$ itself a scroll and $B$ a fibre, 2) $(X,L)=(P^{2}\times P^{2},\mathscr{O}_{P^{2}\times P^{2}}(1,1))$ with $A\in|\mathscr{O}_{P^{2}\times P^{2}}(1,0)|,$ $B\in|\mathscr{O}_{P^{2}\times P^{2}}(0,1)|,$ $3)X=P_{P^{2}}(\mathscr{V})$ where $\mathscr{V}=\mathscr{O}_{P^{2}}(1)^{\oplus 2}\oplus \mathscr{O}_{P^{2}}(2)$ , $L$ is the tautological line bundle, $A=P_{P^{2}}$ $(\mathscr{O}_{P^{2}}(1)^{\oplus 2}$ , and $ B\in\pi^*$ $|\mathscr{O}_{P^{2}}(2)|$ , where $\pi$ : $X\rightarrow P^{2}$ is the scroll projection. This supplements a recent result of Chandler, Howard, and Sommese.
Publié le : 2001-07-15
Classification:  4-folds,  Hyperplane sections,  Adjunction theory,  Simple normal crossing divisors,  Fano manifolds,  14J35,  14J45,  14C20,  14N30
@article{1213023723,
     author = {LANTERI, Antonio and L. TIRONI, Andrea},
     title = {On reducible hyperplane sections of 4-folds},
     journal = {J. Math. Soc. Japan},
     volume = {53},
     number = {3},
     year = {2001},
     pages = { 559-563},
     language = {en},
     url = {http://dml.mathdoc.fr/item/1213023723}
}
LANTERI, Antonio; L. TIRONI, Andrea. On reducible hyperplane sections of 4-folds. J. Math. Soc. Japan, Tome 53 (2001) no. 3, pp.  559-563. http://gdmltest.u-ga.fr/item/1213023723/