The Torelli problem for Logarithmic bundles of hypersurface arrangements in the projective space

Abstract

Let $\mathcal{D}=\{D_1,\dots ,D_{\ell }\}$ be an arrangement of smooth hypersurfaces with normal crossings on the complex projective space ${\mathbb{P}}^n$ and let ${\Omega }_{{\mathbb{P}}^n}^1(log\mathcal{D})$ be the logarithmic bundle attached to it. Our aim is to study the injectivity of the correspondence $\mathcal{D}⟶{\Omega }_{{\mathbb{P}}^n}^1(log\mathcal{D})$. In order to do that, we first show that ${\Omega }_{{\mathbb{P}}^n}^1(log\mathcal{D})$ admits a resolution of length $1$ depending on the degrees and on the equations of $D_1,\dots ,D_{\ell }$. Then, we prove a Torelli type theorem when $\mathcal{D}$ has a sufficiently large number of components of the same degree $d$, by recovering them as unstable smooth irreducible degree-$d$ hypersurfaces of ${\Omega }_{{\mathbb{P}}^n}^1(log\mathcal{D})$. The cases of one quadric and a pair of quadrics in ${\mathbb{P}}^n$ are not Torelli; in particular, through a duality argument, we prove that the isomorphism class of the logarithmic bundle attached to a pair of quadrics is determined by the tangent hyperplanes to the pair. Finally, by describing the moduli spaces containing ${\Omega }_{{\mathbb{P}}^2}^1(log\mathcal{D})$, we show that some line-conic arrangements are not of Torelli type.

99 pages, 13 figures, Ph. D. thesis defended at Università di Firenze (Italy) on 6 May 2013. Related papers: arXiv 1410.8770 and 1304.5709 [math.AG]

References in corpus (1)

• Logarithmic vector fields along smooth plane cubic curves

Cited by in corpus (1)

• Logarithmic bundles of multi-degree arrangements in ${\mathbf{P}}^n$