Poincaré bundles for projective surfaces (Q788777)

Let X be a smooth projective surface, K the canonical divisor, H a very ample divisor and \(M_ H(c_ 1,c_ 2)\) the moduli space of rank-two vector bundles, H-stable with Chern classes \(c_ 1\) and \(c_ 2\). We prove that, if there exists c'\({}_ 1\) such that \(c_ 1\) is numerically equivalent to 2c'\({}_ 1\) and if \(c_ 2-\frac{1}{4}c^ 2_ 1\) is even, greater or equal to \(H^ 2+HK+4\), then there is no Poincaré bundle on \(M_ H(c_ 1,c_ 2)\times X\). Conversely, if there exists c'\({}_ 1\) such that the number c'\({}_ 1\cdot c_ 1\) is odd or if \(\frac{1}{2}c^ 2_ 1-\frac{1}{2}c_ 1\cdot K-c_ 2\) is odd, then there exists a Poincaré bundle on \(M_ H(c_ 1,c_ 2)\times X\).