Summary |
In this report, concepts of wall and chamber are generalized from rank 2 to higher rank. The generalized wall is defined by explicit requirements similar to the rank 2 situation. It is proved that semistability condition with respect to ample divisors in the same chamber are the same. It is also discovered that S-equivalence is independent of ample divisors in chambers. The criteria for the existence of sheaves with different slope stabilities with respect to ample divisors in adjacent chambers are given when the rank is 2 or 3. Suppose W is a wall separating adjacent chambers C- and C+ and ε is a sheaf which is L- semistable for L- in C- and is L+ unstable for L+ in C+. It is shown that with respect to any ample divisors in the associated chamber in C+ next to W, the Harder-Narasimhan filtrations are the same and all its terms determine W. Changes of Harder-Narasimhan filtration as L+ moves away from W is studied. The walls of ruled surfaces is studied and a way to represent walls as an ordered pair satisfying some requirements is introduced. Using this representation, the height and properties of a bunch of consecutive walls starting from the highest one are determined in the following cases: (1) c1 = O(mσ + nf) and gcd(r,m) = 1, (2) rank equal to 2, (3) rank equal to 3, c1 = O(mσ + nf) and 3∤m. When r = 2 and S ≠ P1 x P1, it is proved that all walls separate μ-stability. Also, when Δ( 2, O(2kσ + nf) , c2) > 0 or g > 0 where n is an odd integer, a sheaf is constructed which is μ-stable with respect to all ample divisors aσ + bf where b/a > min{0,(e + 1) / 2} . It is showed that when rank is 3 and c1 = O(mσ + nf) with 3∤m, ML(3, c1, c2) is empty for L in "many" high chambers. The highest chamber C' where ML(3, c1, c2) is non-empty for L ∈ C' when Δ(3, c1, c2) > max{2e, 0} and g > 0 is explicitly determined. Under the same assumption, it is proved that with respect to L in C', all semistable sheaves are locally free. |