These results hold both in characteristic 0Īnd in characteristic p, for all primes p. Of the Grothendieck-Knudsen moduli space M 0,n of stable rational curves is not polyhedralįor n≥10. We construct projective toric surfaces whose blow-upĪs a consequence, we prove that the pseudo-effective cone
Lecture video (Seminario de Geometria Algebraica, Universidad Catolica de Chile) Our proof is based on an analysis of wall-crossing between moduli spaces of stable pairs,Ĭombining classical vector bundles techniques with the method of windows.īlown-up toric surfaces with non-polyhedral effective cone, We conjecture that the subcategory generated by our blocks has a trivial semi-orthogonal complement, It has two blocks for each i-th symmetric power of C for i=0.,g-2 and one block for the (g-1)-st symmetric power. WeĬonstruct a semi-orthogonal decomposition of the bounded derived category of N conjectured by Narasimhan and by Belmans, Galkin and Mukhopadhyay. Let C be a smooth projective curve of genus g>1 and let N be the moduli space of stable rank 2 vector bundles on C of odd degree. For example, we describe strong exceptional collections of length 10 on all Dolgachev surfaces and classify admissible embeddings of derived categories of quivers into derived categories of rational surfaces. We show that they are Morita-equivalent to path algebras R̂ of acyclic quivers for general smoothings within each irreducible component of the versal deformation space of W (as described by Kollár and Shepherd-Barron).įurthermore, R̂ is semi-simple if and only if the smoothing is Q-Gorenstein (one direction is due to Kawamata). We study the deformations of the algebra R induced by the deformations of the surface W to a smooth surface. It is well known that a 2-dimensional cyclic quotient singularity W has the same singularity category as a finite dimensional associative algebra R introduced by Kalck and Karmazyn. In this work, we employ the theory of weaving patterns to eliminate the possibility of a phantom,Ĭompleting the proof of the decompositionĬonjectured by Narasimhan and, independently, by Belmans, Galkin, and Mukhopadhyay.Ĭategorical aspects of the Kollár-Shepherd-Barron correspondence, Into bounded derived categories of symmetric powers of the curve and, possibly,Ī phantom block. With Sebastian Torres, we previously found a semi-orthogonal decomposition of
Let N be the moduli space of stable rank 2 vector bundles on a smooth projective curve of genus g>1 Preprints, papers and books on Algebraic Geometryīraid and Phantom, 39p. Informal description of research interests Latin American School of Algebraic Geometry and Applications, DMS-1935081 Geometry of Compact Moduli Spaces, DMS-0701191 (and REU supplement DMS-0827994) Geometry of Moduli Spaces of Curves and Surfaces, DMS-1001344 Moduli Spaces of Curves and Surfaces, DMS-1303415 (and REU supplement DMS-1539510) Moduli Spaces: New Directions, DMS-1701704 New Frontiers of Algebraic Geometry, DMS-2101726
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