Line Bundles and Curves on a
نویسنده
چکیده
Orders on surfaces provided a rich source of examples of noncommutative surfaces. In [HS05] the authors prove the existence of the analogue of the Picard scheme for orders and in [CK11] the Picard scheme is explicitly computed for an order on P ramified on a smooth quartic. In this paper, we continue this line of work, by studying the Picard and Hilbert schemes for an order on P ramified on a union of two conics. Our main result is that, upon carefully selecting the right Chern classes, the Hilbert scheme is a ruled surface over a genus two curve. Furthermore, this genus two curve is, in itself, the Picard scheme of the order.
منابع مشابه
Moduli of Roots of Line Bundles on Curves
We treat the problem of completing the moduli space for roots of line bundles on curves. Special attention is devoted to higher spin curves within the universal Picard scheme. Two new different constructions, both using line bundles on nodal curves as boundary points, are carried out and compared with pre-existing ones.
متن کاملVector Bundles on Riemann Surfaces
1. Differentiable Manifolds 2 2. Complex Manifolds 3 2.1. Riemann Surfaces of Genus One 4 2.2. Constructing Riemann Surfaces as Curves in P 6 2.3. Constructing Riemann Surfaces as Covers 9 2.4. Constructing Riemann Surfaces by Glueing 10 3. Topological Vector Bundles 11 3.1. The Tangent and Cotangent Bundles 13 3.2. Interlude: Categories, Complexes and Exact Sequences 14 3.3. Metrics on Vector ...
متن کاملLinear Series on Semistable Curves
We study h 0 (X, L) for line bundles L on a semistable curve X of genus g, parametrized by the compactified Picard scheme. The theorem of Riemann is shown to hold. The theorem of Clifford is shown to hold in the following cases: X has two components; X is any semistable curve and d = 0 or d = 2g − 2; X is stable, free from separating nodes, and d ≤ 4. These results are shown to be sharp. Applic...
متن کاملMt822: Introduction to Algebraic Geometry
1. Algebraic varieties 2 1.1. Affine varieties 2 1.2. Projective varieties 2 1.3. Zariski topology 3 1.4. Algebraic geometry and analytic geometry 3 1.5. Singular varieties 3 1.6. Ideals 4 1.7. Regular functions and maps 5 2. Sheaves and cohomology 6 2.1. The Mittag-Leffler problem 7 2.2. Sheaves 7 2.3. Maps of sheaves 8 2.4. Stalks and germs 10 2.5. Cohomology of sheaves 11 3. Vector bundles, ...
متن کاملLectures on Principal Bundles
The aim of these lectures is to give a brief introduction to principal bundles on algebraic curves towards the construction of the moduli spaces of semistable principal bundles. The first lecture develops the basic machinery on principal bundles, their automorphisms. At the end of the first chapter, we give a proof of theorem of Grothendieck on orthogonal bundles. The second chapter, after deve...
متن کاملSmall rational curves on the moduli space of stable bundles
For a smooth projective curve C with genus g ≥ 2 and a degree 1 line bundle L on C, let M := SUC(r,L) be the moduli space of stable vector bundles of rank r over C with the fixed determinant L. In this paper, we study the small rational curves on M and estimate the codimension of the locus of the small rational curves. In particular, we determine all small rational curves when r = 3.
متن کامل