Moduli spaces of quasitrivial sheaves on the three dimensional projective space

Abstract

A torsion free sheaf E on P d is called quasitrivial if E ∨∨ = O ⊕r P 3 and dim(E ∨∨ /E) = 0. While such sheaves are always µ-semistable, they may not be semistable. We study the Gieseker-Maruyama moduli space N (r, n) of rank r semistable quasitrivial sheaves on P 3 with h 0 (E ∨∨ /E) = n via the Quot scheme of points Quot(O ⊕r P 3 , n). We show that N (r, n) is empty when r > n, while N (n, n) has no stable points and is isomorphic to the symmetric product Sym n (P 3 ). Our main result is the construction of an irreducible component of N (r, n) of dimension 2n+rn−r 2 +1 when r < n. Furthermore, this is the only irreducible component when n ≤ 10.