Interpolation and moduli spaces of vector bundles on very general blowups of the projective plane (Q6566553)

In this extremely nicely written paper the authors study certain moduli spaces of vector bundles on blowups of \(\mathbb{P}^{2}_{\mathbb{C}}\) in at least ten very general points. More precisely, let \(X\) be the blowup of \(\mathbb{P}^{2}_{\mathbb{C}}\) at \(n\) very general points \(p_{1}, \dots,p_{n}\) and let us denote by \(H\) the pullback of the class of a line. Denote by \(E_{i}\) the exceptional divisors and let \(E = \sum_{i=1}^{n}E_{i}\). We polarize the surface \(X\) by an ample divisor of the form \(A_{t} = tH-E\). Denote by \(K=K_{X} = -3K+E\) and \(B =\sqrt{n}H-E\). We let \(M_{X,A_{t}}(r,c_{1}, \chi )\) denote the moduli space of \(A_{t}\)-semistable sheaves with the numerical invariants \((r,c_{1}, \chi )\), where \(r\) denotes the rank, \(c_{1}\) is the first Chern class, and \(\chi\) is the Euler characteristic. The first main result of the paper can be formulated as follows (in these cases the Nagata conjecture holds).\N\NTheorem A. Let \(n=16\). For \(14/3 < t < 16/3\) the moduli space \(M_{X,A_{t}}(2,K,2)\) is isomorphic to \(\mathbb{P}^{5}\). For \(4 < t < 14/3\) the moduli space \(M_{X,A_{t}}(2,K,2)\) is isomorphic to a blowup of \(\mathbb{P}^{5}\) along \(16\) points.\N\NLet \(n=25\). For \(5 < t \leq 27/5\) the moduli space \(M_{X,A_{t}}(2,K,4)\) is isomorphic to a disjoint union of \(25\) copies of \(\mathbb{P}^{8}\).\N\NTheorem B. Let \(10 \leq n \leq 15\) and assume that the SHGH conjecture holds.\N\N1) If \(t > n/3\), then \(M_{X,A_{t}}(2,K,2)\) is empty.\N\N2) Suppose that \(11 \leq n \leq 15\). As \(t\) decreases past \(n/3\), \(M_{X,A_{t}}(2,K,2)\) acquires a component isomorphic to \(\mathbb{P}^{n-11}\). For \(11 \leq n \leq 12\), this component persists without modification as \(t\) decreases to \(\sqrt{n}\). For \(13 \leq n \leq 15\), this component is blown up at \(n\) points as \(t\) decreases past \((n-2)/3\) and then persists without modification as \(t\) decreases to \(\sqrt{n}\).\N\N3) For every non-trivial non-exceptional divisor \(D\) satisfying \(\chi(D) \geq 1\) and \(2B\cdot D < B \cdot K\), the moduli space \(M_{X,A_{t}}(2,K,2)\) acquires a new component isomorphic to \(\mathbb{P}^{-\chi(2D-K)-1}\) as \(t\) decreases past \(t_{D}\) - here \(t_{D}\) is a unique value such that \(2A_{t_{D}}\cdot D = A_{t_{D}}\cdot K\).\N\N4) There is a complete description of the components \(M_{X,A_{t}}(2,K,2)\), and all they are disjoint.