Effective divisors on the Hilbert scheme of points in the plane and interpolation for stable bundles (Q2788618)

The main work of this paper is to determine the full cone \(\text{Eff}\mathbb{P}^{2[n]}\) of effective divisors on the Hilbert scheme \(\mathbb{P}^{2[n]}\) of \(n\) points over \(\mathbb{P}^2\).NEWLINENEWLINEThe Picard group of \(\mathbb{P}^{2[n]}\) has rank 2 and is generated by \(H\) and \(\Delta/2\), where \(H\) is given by the \(\mathscr{S}_n\)-linearized line bundle \(\mathcal{O}_{\mathbb{P}^2}(1)^{\boxtimes n}\) over \((\mathbb{P}^2)^n\), and \(\Delta\) is the locus of nonreduced schemes, also \(\Delta/2=c_1(\det(R^{\bullet}p(q^*\mathcal{O}_{\mathbb{P}^2}\otimes I_{\mathcal{Z}}))^{\vee})\), with \(q:\mathbb{P}^2\times\mathbb{P}^{2[n]}\rightarrow \mathbb{P}^2\), \(p:\mathbb{P}^2\times\mathbb{P}^{2[n]}\rightarrow \mathbb{P}^{2[n]}\) two projections, and \(I_{\mathcal{Z}}\) the ideal sheaf of the universal family \(\mathcal{Z}\subset \mathbb{P}^2\times\mathbb{P}^{2[n]}\).NEWLINENEWLINEIt is well-known that \(\Delta\) always spans on edge of the cone \(\text{Eff }\mathbb{P}^{2[n]}\). Also if \(n=\binom{r+2}{2}\), using Gaeta's resolution one can show easily that the other edge of \(\text{Eff }\mathbb{P}^{2[n]}\) is spanned by \(rH-\Delta/2=c_1(\det(R^{\bullet}p(q^*\mathcal{O}_{\mathbb{P}^2}(r)\otimes I_{\mathcal{Z}}))^{\vee})\). For a general \(n\), the author generalized Gaeta's resolution by replacing \(\mathcal{O}_{\mathbb{P}^2}(-r)^{\oplus kn}\) by some semiexceptional bundles. After a number of calculations and analysis on the properties of exceptional bundles, the author finally obtained that for every \(n\in\mathbb{Z}_{+}\), there is a stable bundle \(V\) such that \(\chi(V)/r(V)=n\) and its slop \(\mu(V)>0\) is minimal among all stable bundles \(V'\) with property \(\chi(V')/r(V')\geq n\) and \(\mu(V')>0\). So \(\chi(V\otimes I_{Z})=0\) for every \(Z\in\mathbb{P}^{2[n]}\). Applying some theories of Kronecker modules, the author proved that \(H^i(V\otimes I_Z)=0,~i=0,1,2\) for a general \(Z\). Hence the set \(D_V(n):=\{Z\in\mathbb{P}^{2[n]}\big| H^1(V\otimes I_Z)\neq 0.\}\) is a divisor of class \(\mu(V)H-\Delta/2\) on \(\mathbb{P}^{2[n]}\). The last thing is to check \(D_V(n)\) is extremal. It is enough to construct a complete curve \(C\subset \mathbb{P}^{2[n]}\) such that \(C\cap D_V(n)=\emptyset\). The existence of such curve \(C\) follows from the fact that general \(I_Z\) can be resolved by two fixed bundles \(E_1,E_2\) as follows NEWLINE\[NEWLINE0\rightarrow E_1\rightarrow{\phi} E_2\rightarrow I_Z\rightarrow 0,NEWLINE\]NEWLINE and the maps failing to give torsion-free cokernels form a subset of codimension \(\geq 2\) inside \(\text{Hom}(E_1,E_2)\).NEWLINENEWLINEAt last, the author discussed the Bridgeland stability of \(I_Z\) and showed that the collapsing wall for \(\mathbb{P}^{2[n]}\) corresponds exactly to the nontrivial edge \(\mu H-\Delta/2\), which provides an important evidence to the conjecture asserting the correspondence between Bridgeland and Mori walls for moduli spaces of semistable sheaves over \(\mathbb{P}^2\).