Let D = {D_1, . . . , D_l} be an arrangement of smooth hypersurfaces with normal crossings on the complex projective space P^n and let Omega^1_P^n(logD) be the logarithmic bundle attached to it. Following (Ancona in Notes of a talk given in Florence, 1998), we show that Omega^1 _P^n(logD) admits a resolution of length 1 which explicitly depends on the degrees and on the equations of D_1, . . . , D_l. Then we prove a Torelli type theorem when all the D_i ’s have the same degree d and l ≥ (n+d d) + 3: indeed, we recover the components of D as unstable smooth hypersurfaces of Omega^1_P^n(logD). Finally we analyze the cases of one quadric and a pair of quadrics, which yield examples of non-Torelli arrangements. In particular, through a duality argument, we prove that two pairs of quadrics have isomorphic logarithmic bundles if and only if they have the same tangent hyperplanes.
Logarithmic bundles of hypersurface arrangements in Pn
ANGELINI, Elena
2014
Abstract
Let D = {D_1, . . . , D_l} be an arrangement of smooth hypersurfaces with normal crossings on the complex projective space P^n and let Omega^1_P^n(logD) be the logarithmic bundle attached to it. Following (Ancona in Notes of a talk given in Florence, 1998), we show that Omega^1 _P^n(logD) admits a resolution of length 1 which explicitly depends on the degrees and on the equations of D_1, . . . , D_l. Then we prove a Torelli type theorem when all the D_i ’s have the same degree d and l ≥ (n+d d) + 3: indeed, we recover the components of D as unstable smooth hypersurfaces of Omega^1_P^n(logD). Finally we analyze the cases of one quadric and a pair of quadrics, which yield examples of non-Torelli arrangements. In particular, through a duality argument, we prove that two pairs of quadrics have isomorphic logarithmic bundles if and only if they have the same tangent hyperplanes.I documenti in SFERA sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.
