scholarly journals Local BPS Invariants: Enumerative Aspects and Wall-Crossing

2018 ◽  
Vol 2020 (17) ◽  
pp. 5450-5475 ◽  
Author(s):  
Jinwon Choi ◽  
Michel van Garrel ◽  
Sheldon Katz ◽  
Nobuyoshi Takahashi
Keyword(s):  
Projective Space ◽  
Euler Characteristic ◽  
Moduli Spaces ◽  
Del Pezzo Surfaces ◽  
Arithmetic Genus ◽  
One Dimensional ◽  
Wall Crossing ◽  
Bps Invariants ◽  
Del Pezzo ◽  
Dimensional Projective Space

Abstract We study the BPS invariants for local del Pezzo surfaces, which can be obtained as the signed Euler characteristic of the moduli spaces of stable one-dimensional sheaves on the surface $S$. We calculate the Poincaré polynomials of the moduli spaces for the curve classes $\beta $ having arithmetic genus at most 2. We formulate a conjecture that these Poincaré polynomials are divisible by the Poincaré polynomials of $((-K_S).\beta -1)$-dimensional projective space. This conjecture motivates the upcoming work on log BPS numbers [8].

scholarly journals Stable pair, tropical, and log canonical compactifications of moduli spaces of del Pezzo surfaces

2009 ◽  
Vol 178 (1) ◽  
pp. 173-227 ◽  
Author(s):  
Paul Hacking ◽  
Sean Keel ◽  
Jenia Tevelev
Keyword(s):  
Moduli Spaces ◽  
Del Pezzo Surfaces ◽  
Stable Pair ◽  
Log Canonical ◽  
Del Pezzo

scholarly journals Derived categories of quintic del Pezzo fibrations

2021 ◽  
Vol 27 (1) ◽  
Author(s):  
Fei Xie
Keyword(s):  
Moduli Space ◽  
Moduli Spaces ◽  
Derived Categories ◽  
Derived Category ◽  
The Other ◽  
Del Pezzo Surfaces ◽  
Linear Section ◽  
Projective Duality ◽  
Del Pezzo ◽  
Space Approach

AbstractWe provide a semiorthogonal decomposition for the derived category of fibrations of quintic del Pezzo surfaces with rational Gorenstein singularities. There are three components, two of which are equivalent to the derived categories of the base and the remaining non-trivial component is equivalent to the derived category of a flat and finite of degree 5 scheme over the base. We introduce two methods for the construction of the decomposition. One is the moduli space approach following the work of Kuznetsov on the sextic del Pezzo fibrations and the components are given by the derived categories of fine relative moduli spaces. The other approach is that one can realize the fibration as a linear section of a Grassmannian bundle and apply homological projective duality.

scholarly journals DERIVED SCHWARZ MAP OF THE HYPERGEOMETRIC DIFFERENTIAL EQUATION AND A PARALLEL FAMILY OF FLAT FRONTS

2008 ◽  
Vol 19 (07) ◽  
pp. 847-863 ◽  
Author(s):  
TAKESHI SASAKI ◽  
KOTARO YAMADA ◽  
MASAAKI YOSHIDA
Keyword(s):  
Differential Equation ◽  
Projective Space ◽  
Hyperbolic Space ◽  
Three Dimensional ◽  
One Dimensional ◽  
Flat Surfaces ◽  
Hypergeometric Differential Equation ◽  
The One ◽  
Dimensional Projective Space

In one of the previous papers, we defined a map, called the hyperbolic Schwarz map, from the one-dimensional projective space to the three-dimensional hyperbolic space by use of solutions of the hypergeometric differential equation, and thus obtained closed flat surfaces belonging to the class of flat fronts. We continue the study of such flat fronts in this paper. First, we introduce derived Schwarz maps of the hypergeometric differential equation and, second, we construct a parallel family of flat fronts connecting the classical Schwarz map and the derived Schwarz map.

scholarly journals Tropical Lagrangians in toric del-Pezzo surfaces

2021 ◽  
Vol 27 (1) ◽  
Author(s):  
Jeffrey Hicks
Keyword(s):  
Lagrangian Submanifold ◽  
Del Pezzo Surfaces ◽  
Dimer Model ◽  
Mirror Pair ◽  
Wall Crossing ◽  
Lagrangian Tori ◽  
Anticanonical Divisor ◽  
Del Pezzo

AbstractWe look at how one can construct from the data of a dimer model a Lagrangian submanifold in $$(\mathbb {C}^*)^n$$ ( C ∗ ) n whose valuation projection approximates a tropical hypersurface. Each face of the dimer corresponds to a Lagrangian disk with boundary on our tropical Lagrangian submanifold, forming a Lagrangian mutation seed. Using this we find tropical Lagrangian tori $$L_{T^2}$$ L T 2 in the complement of a smooth anticanonical divisor of a toric del-Pezzo whose wall-crossing transformations match those of monotone SYZ fibers. An example is worked out for the mirror pair $$(\mathbb {CP}^2{\setminus } E, W), {\check{X}}_{9111}$$ ( CP 2 \ E , W ) , X ˇ 9111 . We find a symplectomorphism of $$\mathbb {CP}^2{\setminus } E$$ CP 2 \ E interchanging $$L_{T^2}$$ L T 2 and a SYZ fiber. Evidence is provided that this symplectomorphism is mirror to fiberwise Fourier–Mukai transform on $${\check{X}}_{9111}$$ X ˇ 9111 .

scholarly journals Change of Polarization for Moduli of Sheaves on Surfaces as Bridgeland Wall-crossing

2018 ◽  
Vol 2020 (7) ◽  
pp. 2007-2033
Author(s):  
Aaron Bertram ◽  
Cristian Martinez
Keyword(s):  
Moduli Spaces ◽  
Stability Conditions ◽  
One Dimensional ◽  
Wall Crossing ◽  
Moduli Of Sheaves ◽  
Twisted Sheaves

Abstract We prove that the “Thaddeus flips” of L-twisted sheaves constructed by Matsuki and Wentworth explaining the change of polarization for Gieseker semistable sheaves on a surface can be obtained via Bridgeland wall-crossing. Similarly, we realize the change of polarization for moduli spaces of one-dimensional Gieseker semistable sheaves on a surface by varying a family of stability conditions.

scholarly journals Compact moduli spaces of Del Pezzo surfaces and Kähler–Einstein metrics

2016 ◽  
Vol 102 (1) ◽  
pp. 127-172 ◽  
Author(s):  
Yuji Odaka ◽  
Cristiano Spotti ◽  
Song Sun
Keyword(s):  
Moduli Spaces ◽  
Einstein Metrics ◽  
Del Pezzo Surfaces ◽  
Del Pezzo ◽  
Kähler Einstein Metrics
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