9,534 research outputs found
The Picard Group of Brauer-Severi Varieties
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
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
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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