Stability and moduli spaces of syzygy bundles

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It is a longstanding problem in Algebraic Geometry to determine whether the syzygy bundle Ed₁,...,dₙ on Pⁿ defined as the kernel of a general epimorphism ϕ:O(-d₁)⊕...(-dₙ) is (semi)stable. In this thesis, attention is restricted to the case of syzygy bundles Syz(f₁,...,fₙ) on Pⁿ associated to n generic forms f₁,...,fₙ∈ K[X₀,...,Xₙ] of the same degree d, for N≥2. The first goal is to prove that Syz(f₁,...,fₙ) is stable if N+1≤ n≤d+NN, except for the case (N,n,d)=(2,5,2). The second is to study moduli spaces of stable rank n-1 vector bundles on Pⁿ containing syzygy bundles. In a joint work with Laura Costa and Rosa Marıa Miró-Roig, we prove that N, d and n are as above, then the syzygy bundle Syz(f₁,...,fₙ) is unobstructed and it belongs to a generically smooth irreducible component of dimension nd+NN-n², if N≥3, and nd+22+nd-12-n², if N=2. The results in chapter 3, for N≥3, were obtained independently by Iustin Coandă in arXiv:0909.4435.