scholarly journals Virtual Poincaré polynomial of the space of stable pairs supported on quintic curves

2015 ◽  
Vol 94 ◽  
pp. 209-217
Author(s):  
Kiryong Chung
Keyword(s):  
Poincaré Polynomial ◽  
Virtual Poincaré Polynomial ◽  
Quintic Curves ◽  
Stable Pairs

scholarly journals AN INVERSE MAPPING THEOREM FOR BLOW-NASH MAPS ON SINGULAR SPACES

2016 ◽  
Vol 223 (1) ◽  
pp. 162-194 ◽  
Author(s):  
JEAN-BAPTISTE CAMPESATO
Keyword(s):  
Jacobian Determinant ◽  
Motivic Integration ◽  
Inverse Mapping ◽  
Singular Spaces ◽  
Poincaré Polynomial ◽  
Algebraic Sets ◽  
Change Of Variables ◽  
Inverse Mapping Theorem ◽  
Virtual Poincaré Polynomial ◽  
Real Analytic

A semialgebraic map $f:X\rightarrow Y$ between two real algebraic sets is called blow-Nash if it can be made Nash (i.e., semialgebraic and real analytic) by composing with finitely many blowings-up with nonsingular centers.We prove that if a blow-Nash self-homeomorphism $f:X\rightarrow X$ satisfies a lower bound of the Jacobian determinant condition then $f^{-1}$ is also blow-Nash and satisfies the same condition.The proof relies on motivic integration arguments and on the virtual Poincaré polynomial of McCrory–Parusiński and Fichou. In particular, we need to generalize Denef–Loeser change of variables key lemma to maps that are generically one-to-one and not merely birational.

scholarly journals Polynomial identities related to special Schubert varieties

2021 ◽  
Author(s):  
Francesca Cioffi ◽  
Davide Franco ◽  
Carmine Sessa
Keyword(s):  
Arbitrary Number ◽  
Schubert Variety ◽  
Decomposition Theorem ◽  
Point Of View ◽  
Explicit Description ◽  
Intersection Cohomology ◽  
Schubert Varieties ◽  
Polynomial Identities ◽  
Poincaré Polynomial

AbstractLet $$\mathcal S$$ S be a single condition Schubert variety with an arbitrary number of strata. Recently, an explicit description of the summands involved in the decomposition theorem applied to such a variety has been obtained in a paper of the second author. Starting from this result, we provide an explicit description of the Poincaré polynomial of the intersection cohomology of $$\mathcal S$$ S by means of the Poincaré polynomials of its strata, obtaining interesting polynomial identities relating Poincaré polynomials of several Grassmannians, both by a local and by a global point of view. We also present a symbolic study of a particular case of these identities.

scholarly journals Poincaré polynomials of generic torus orbit closures in Schubert varieties

2021 ◽  
pp. 189-208
Author(s):  
Eunjeong Lee ◽  
Mikiya Masuda ◽  
Seonjeong Park ◽  
Jongbaek Song
Keyword(s):  
Schubert Variety ◽  
Flag Variety ◽  
Schubert Varieties ◽  
Orbit Closure ◽  
Poincaré Polynomial ◽  
Content Type ◽  
Eulerian Polynomial ◽  
Orbit Closures

The closure of a generic torus orbit in the flag variety G / B G/B of type  A A is known to be a permutohedral variety, and its Poincaré polynomial agrees with the Eulerian polynomial. In this paper, we study the Poincaré polynomial of a generic torus orbit closure in a Schubert variety in  G / B G/B . When the generic torus orbit closure in a Schubert variety is smooth, its Poincaré polynomial is known to agree with a certain generalization of the Eulerian polynomial. We extend this result to an arbitrary generic torus orbit closure which is not necessarily smooth.

scholarly journals Curve counting via stable pairs in the derived category

2009 ◽  
Vol 178 (2) ◽  
pp. 407-447 ◽  
Author(s):  
R. Pandharipande ◽  
R. P. Thomas
Keyword(s):  
Derived Category ◽  
Stable Pairs
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