9,534 research outputs found

    The Picard Group of Brauer-Severi Varieties

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    In this note we provide explicit generators of the Picard groups of cyclic Brauer-Severi varieties defined over the base field. In particular, for all Brauer-Severi surfaces. To produce these generators we use the Twisting Theory for smooth plane curves

    The canonical pencils on Horikawa surfaces

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    We calculate the monodromies of the canonical Lefschetz pencils on a pair of homeomorphic Horikawa surfaces. We show in particular that the (pluri)canonical pencils on these surfaces have the same monodromy groups, and are related by a "partial twisting" operation.Comment: This is the version published by Geometry & Topology on 29 November 200

    Hilbert schemes of points on a locally planar curve and the Severi strata of its versal deformation

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    Let C be a locally planar curve. Its versal deformation admits a stratification by the genera of the fibres. The strata are singular; we show that their multiplicities at the central point are determined by the Euler numbers of the Hilbert schemes of points on C. These Euler numbers have made two prior appearances. First, in certain simple cases, they control the contribution of C to the Pandharipande-Thomas curve counting invariants of three-folds. In this context, our result identifies the strata multiplicities as the local contributions to the Gopakumar-Vafa BPS invariants. Second, when C is smooth away from a unique singular point, a special case of a conjecture of Oblomkov and Shende identifies the Euler numbers of the Hilbert schemes with the "U(infinity)" invariant of the link of the singularity. We make contact with combinatorial ideas of Jaeger, and suggest an approach to the conjecture.Comment: 16 page
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