Author
I. Dolgachev
Bio: I. Dolgachev is an academic researcher. The author has contributed to research in topics: Arrangement of hyperplanes & Splitting principle. The author has an hindex of 1, co-authored 1 publications receiving 103 citations.
Papers
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TL;DR: In this paper, the authors study the bundles of logarithmic 1-forms corresponding to such divisors from the point of view of classification of vector bundles on $P^n.
Abstract: Any arrangement of hyperplanes in general position in $P^n$ can be regarded as a divisor with normal crossing. We study the bundles of logarithmic 1-forms corresponding to such divisors` from the point of view of classification of vector bundles on $P^n$. It turns out that all such bundles are stable. The study of jumping lines of these bundles gives a unified treatment of several classical constructions associating a curve to a collection of points in $P^n$. The main result of the paper is \"Torelli theorem\" which says that the collection of hyperplanes can be recovered from the isomorphism class of the corresponding logarithmic bundle unless the hyperplanes ocsulate a rational normal curve. In this latter case our construction reduces to that of secant bundles of Schwarzenberger.
110 citations
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Book•
08 Oct 2012TL;DR: In this paper, the authors present a survey of the geometry of lines and cubic surfaces, including determinantal equations, theta characteristics, and the Cremona transformations.
Abstract: Preface 1. Polarity 2. Conics and quadrics 3. Plane cubics 4. Determinantal equations 5. Theta characteristics 6. Plane quartics 7. Cremona transformations 8. Del Pezzo surfaces 9. Cubic surfaces 10. Geometry of lines Bibliography Index.
663 citations
114 citations
TL;DR: In this article, a Hormander-Mihlin multiplier theorem for finite-dimensional cocycles with optimal smoothness condition was proved for groups of arbitrary discrete groups and a non-commutative generalization of Calderon-Zygmund theory was proposed.
Abstract: We investigate Fourier multipliers on the compact dual of arbitrary discrete groups. Our main result is a Hormander–Mihlin multiplier theorem for finite-dimensional cocycles with optimal smoothness condition. We also find Littlewood–Paley type inequalities in group von Neumann algebras, prove L
p
estimates for noncommutative Riesz transforms and characterize L
∞ → BMO boundedness for radial Fourier multipliers. The key novelties of our approach are to exploit group cocycles and cross products in Fourier multiplier theory in conjunction with BMO spaces associated to semigroups of operators and a noncommutative generalization of Calderon–Zygmund theory.
94 citations
TL;DR: The Gale transform, an involution on sets of points in projective space, appears in a multitude of guises and in subjects as diverse as optimization, coding theory, theta functions, and recently in our proof that certain general sets of objects fail to satisfy the minimal free resolution conjecture as mentioned in this paper.
80 citations
TL;DR: For an essential central hyperplane arrangement A ⊆ V ≃ k n+1, this article showed that 1 (A) gives rise to a locally free sheaf on P n if and only if for all X ∈ LA with rank X < dim V, the module 1 (AX) is free.
71 citations