Mathematical instantons with maximal order jumping lines

Abstract

A mathematical instanton bundle on P (over an algebraically closed field) is a rank two vector bundle E on P with c1 = 0 and with H0(E) = H1(E(−2)) = 0. Let c2(E) = n. Then n > 0. A jumping line of E of order a, (a > 0), is a line ` in P on which E splits as O`(−a)⊕O`(a). It is easy to see that the jumping lines of E all have order ≤ n. We will say that E has a maximal order jumping line if it has a jumping line of order n. Our goal is to show that such an E is unobstructed in the moduli space of stable rank two bundles, i.e., H2(E ⊗ E) = 0. The technique can be slightly extended. We show that when c2 = 5, any E with a jumping line of order 4 is unobstructed. We describe at the end how mathematical instantons with maximal order jumping lines arise and estimate the dimension of this particular smooth locus of bundles. It is known that every mathematical instanton bundle on P with c2 ≤ 4 is unobstructed ([L]). In [H], it is shown that bundles built by the Serre construction from the union of n+1 skew lines and the bundles built from elliptic curves in P of degree n + 4 are unobstructed (with c2 = n.) The SU(2)instanton bundles obtained from physics are also known to be unobstructed using analytic arguments ([D-V]). Recently Nusler and Trautmann [N-T] have shown that any mathematical instanton with a section in degree 1 is unobstructed, extending the case of bundles obtained from skew lines and also extending a result of Hirschowitz and Narasimhan on ’t Hooft bundles [H-N]. A preprint of Ancona and Ottaviani ([A-O]) produces a singular point on the moduli space of stable bundles with c1 = 0, c2 = 5. This singular point is in the closure of the open set of mathematical instantons but is not itself a mathematical instanton bundle. Still open is the general Question: for fixed Chern classes c1 = 0, c2 = n, is the moduli space Mmi(0, n) of mathematical instanton bundles on P irreducible and smooth? A construction of stable bundles on P (not necessarily mathematical instanton) with jumping lines of large order was first done by C. Peskine.