Logarithmic bundles of hypersurface arrangements in \mathbf{P}^n

• Autores: Elena Angelini
• Localización: Collectanea mathematica, ISSN 0010-0757, Vol. 65, Fasc. 3, 2014, págs. 285-302
• Idioma: inglés
• DOI: 10.1007/s13348-014-0112-0
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• Resumen

  • Let {\mathcal {D}}=\{D_{1}, \ldots , D_{\ell }\} be an arrangement of smooth hypersurfaces with normal crossings on the complex projective space \mathbf{P}^n and let \Omega ^{1}_{\mathbf{P}^n}(\log {\mathcal {D}}) be the logarithmic bundle attached to it. Following (Ancona in Notes of a talk given in Florence, 1998), we show that \Omega ^{1}_{\mathbf{P}^n}(\log {\mathcal {D}}) admits a resolution of length 1 which explicitly depends on the degrees and on the equations of D_{1},\ldots ,D_{\ell }. Then we prove a Torelli type theorem when all the D_{i}’s have the same degree d and \ell \ge {{n+d}\atopwithdelims (){d}}+3: indeed, we recover the components of {\mathcal {D}} as unstable smooth hypersurfaces of \Omega ^{1}_{\mathbf{P}^n}(\log {\mathcal {D}}). 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.