A note on Bridgeland moduli spaces and moduli spaces of sheaves on $$X_{14}$$ and $$Y_3$$

Abstract

AbstractWe study Bridgeland moduli spaces of semistable objects of $$(-1)$$ ( - 1 ) -classes and $$(-4)$$ ( - 4 ) -classes in the Kuznetsov components on index one prime Fano threefold $$X_{4d+2}$$ X 4 d + 2 of degree $$4d+2$$ 4 d + 2 and index two prime Fano threefold $$Y_d$$ Y d of degree d for $$d=3,4,5$$ d = 3 , 4 , 5 . For every Serre-invariant stability condition on the Kuznetsov components, we show that the moduli spaces of stable objects of $$(-1)$$ ( - 1 ) -classes on $$X_{4d+2}$$ X 4 d + 2 and $$Y_d$$ Y d are isomorphic. We show that moduli spaces of stable objects of $$(-1)$$ ( - 1 ) -classes on $$X_{14}$$ X 14 are realized by Fano surface $$\mathcal {C}(X)$$ C ( X ) of conics, moduli spaces of semistable sheaves $$M_X(2,1,6)$$ M X ( 2 , 1 , 6 ) and $$M_X(2,-1,6)$$ M X ( 2 , - 1 , 6 ) and the correspondent moduli spaces on cubic threefold $$Y_3$$ Y 3 are realized by moduli spaces of stable vector bundles $$M^b_Y(2,1,2)$$ M Y b ( 2 , 1 , 2 ) and $$M^b_Y(2,-1,2)$$ M Y b ( 2 , - 1 , 2 ) . We show that moduli spaces of semistable objects of $$(-4)$$ ( - 4 ) -classes on $$Y_{d}$$ Y d are isomorphic to the moduli spaces of instanton sheaves $$M^{inst}_Y$$ M Y inst when $$d\ne 1,2$$ d ≠ 1 , 2 , and show that there are open immersions of $$M^{inst}_Y$$ M Y inst into moduli spaces of semistable objects of $$(-4)$$ ( - 4 ) -classes when $$d=1,2$$ d = 1 , 2 . Finally, when $$d=3,4,5$$ d = 3 , 4 , 5 we show that these moduli spaces are all isomorphic to $$M^{ss}_X(2,0,4)$$ M X ss ( 2 , 0 , 4 ) .