smooth projective surface
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2020 ◽  
Vol 156 (8) ◽  
pp. 1623-1663
Author(s):  
Amin Gholampour ◽  
Richard P. Thomas

We express nested Hilbert schemes of points and curves on a smooth projective surface as ‘virtual resolutions’ of degeneracy loci of maps of vector bundles on smooth ambient spaces. We show how to modify the resulting obstruction theories to produce the virtual cycles of Vafa–Witten theory and other sheaf-counting problems. The result is an effective way of calculating invariants (VW, SW, local PT and local DT) via Thom–Porteous-like Chern class formulae.



Author(s):  
Yu Zhao

Abstract In this paper, we define the $K$-theoretic Hall algebra for dimension $0$ coherent sheaves on a smooth projective surface, prove that the algebra is associative, and construct a homomorphism to a shuffle algebra introduced by Negut [ 10].



2020 ◽  
Vol 20 (2) ◽  
pp. 233-248
Author(s):  
Andrea Cattaneo

AbstractWe present a way of computing the degree of the secant (resp. tangent) variety of a smooth projective surface, under the assumption that the divisor giving the embedding in the projective space is 3-very ample. This method exploits the link between these varieties and the Hilbert scheme 0-dimensional subschemes of length 2 of the surface.



Author(s):  
Adrian Langer

Abstract We show various properties of smooth projective D-affine varieties. In particular, any smooth projective D-affine variety is algebraically simply connected and its image under a fibration is D-affine. In characteristic 0 such D-affine varieties are also uniruled. We also show that (apart from a few small characteristics) a smooth projective surface is D-affine if and only if it is isomorphic to either ${{\mathbb{P}}}^2$ or ${{\mathbb{P}}}^1\times{{\mathbb{P}}}^1$. In positive characteristic, a basic tool in the proof is a new generalization of Miyaoka’s generic semipositivity theorem.



2019 ◽  
Vol 163 (3-4) ◽  
pp. 361-373
Author(s):  
Roberto Laface ◽  
Piotr Pokora

AbstractIn the present paper we focus on a weighted version of the Bounded Negativity Conjecture, which predicts that for every smooth projective surface in characteristic zero the self-intersection numbers of reduced and irreducible curves are bounded from below by a function depending on the intesection of curve with an arbitrary big and nef line bundle that is positive on the curve. We gather evidence for this conjecture by showing various bounds on the self-intersection number of curves in an algebraic surface. We focus our attention on blow-ups of algebraic surfaces, which have so far been neglected.





2018 ◽  
Vol 20 (07) ◽  
pp. 1750084 ◽  
Author(s):  
Wenfei Liu

Let [Formula: see text] be a minimal smooth projective surface of general type with irregularity [Formula: see text]. We show that, if [Formula: see text] has a nontrivial holomorphic automorphism acting trivially on the cohomology with rational coefficients, then it is a surface isogenous to a product. As a consequence of this geometric characterization, one infers that no nontrivial automorphism of surfaces of general type with [Formula: see text] (which are not necessarily minimal) can be homotopic to the identity. In particular, such surfaces are rigidified in the sense of Fabrizio Catanese.





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