2017/03/20 by Yuan, Yao
#14D05 #14D22 #14J26 #Algebraic Geometry (math.AG) #FOS: Mathematics
paper · doi:10.48550/arxiv.1703.06665
We study Le Potier's strange duality conjecture on a rational surface. We focus on the strange duality map SDcnr,L which involves the moduli space of rank r sheaves with trivial first Chern class and second Chern class n, and the moduli space of 1-dimensional sheaves with determinant L and Euler characteristic 0. We show there is an exact sequence relating the map SDcrr,L to SD_cr-1r,L and SDcrr,L⊗ KX for all r≥1 under some conditions on X and L which applies to a large number of cases on \p2 or Hirzebruch surfaces . Also on ℙ2 we show that for any r>0, SDcrr,dH is an isomorphism for d=1,2, injective for d=3 and moreover SDc33,rH and SDc32,rH are injective. At the end we prove that the map SDcn2,L (n≥2) is an isomorphism for X=ℙ2 or Fano rational ruled surfaces and gL=3, and hence so is SDc33,L as a corollary of our main result.