
## 2. Bridgeland stability

1. #### Problem 2.1.

[Macri, Schmidt] (Macri) Construct Bridgeland stability conditions on degenerate varieties, for example, the hypersurfaces with equations $x_0x_1\dots x_n=0$ in $\mathbb{P}^n$, especially for $n =2,3,4$. Given $[C]\in\overline{\mathcal{M}}_g$ arbitrary, is there a description of the space of stability conditions on $C$? This is known some some irreducible nodal curves.

(Schmidt) Are there Bogomolov type inequalities for stable objects on degenerate varieties?
• #### Problem 2.2.

[Schmidt] Let $X$ be a smooth projective surface. Let $L\in\mathrm{Pic}(X)$ be a globally generated line bundle. Consider the Lazarsfeld-Mukai vector bundle $M_L:=\ker(H^0(X,L)\otimes\mathcal{O}_X\to L)$. For which stability conditions is $M_L$ stable?
• #### Problem 2.3.

[Macri, Farkas] In the same setting as problem 2.2, can one prove property $(N_p)$ for $K_X+nL$ in terms of Bridgeland stability? For $X=C$ a curve, and $\deg L\geq 2g+p+1$, Green in 1984 proved that $(C,L)$ satisfies $(N_p)$. (Kemeny) Can one generalize this to surfaces via Bridgeland stability?
• #### Problem 2.4.

[Schmidt] Let $C \subset \mathbb{P}^3$ be a smooth curve. Is there a characterization for walls in Bridgeland or tilt stability for its ideal sheaf $\mathcal{I}_{C/\mathbb{P}^3}$? It is known that if there exists a wall for tilt stability, then any destablizing sheaf $E\to \mathcal{I}_{C/\mathbb{P}^3}$ must be reflexive. Do such $E$’s admit a stronger characterization?
• #### Problem 2.5.

[Nuer] Given one of the known exceptional collections of stable objects on $\overline{\mathcal{M}}_{0,n}$ (e.g. [MR3098789] or forthcoming work of Castravet-Tevelev), can one use the corresponding Bridgeland stability condition to study birational geometry of $\overline{\mathcal{M}}_{0,n}$ (e.g. its nef or, very greedily, effective cone)?
• #### Problem 2.6.

[Schmidt] Do moduli spaces of Bridgeland stable objects have a projective/good coarse moduli space? The Bayer-Macri divisor is nef. When is it ample? Try the blow-up of $\mathbb{P}^2$ at two points or surfaces of general type.
• #### Problem 2.7.

[Lo] Let $X$ be a smooth surface with $\rho(X)\geq 2$. Vary the polarzation not only along a line but along a plane and describe the structure of walls for tilt stability. A first case to study is that of elliptic fibrations.

Cite this as: AimPL: Stability and moduli spaces, available at http://aimpl.org/stabmoduli.