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 $M_Y^b(2,1,2)$ and $M_Y^b(2,-1,2)$. We show that moduli spaces of semistable objects of $(-4)$-classes on $Y_d$ are isomorphic to the moduli spaces of instanton sheaves $M_Y^{inst}$ when $d\ne 1,2$, and show that there're open immersions of $M_Y^{inst}$ 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 $M_X^{ss}(2,0,4)$.

30 pages. Some typos are fixed. Add the case $d=3$ in Section 9. Comments are welcome

Cited by in corpus (1)

• Serre-invariant stability conditions and Ulrich bundles on cubic threefolds