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

Abstract

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