• Bridgeland
• Geometria Algebrica
• Mukai

Throughout this thesis paper we discuss the notion of stability conditions for triangulated categories introduced by Bridgeland in his PhD thesis, with focus on the applications of Bridgeland’s theory to classical problems in Algebraic Geometry.
In the first part of the survey we will give the definition of a stability condition and we will study the basic geometrical properties of the set of stability conditions. We will show, in particular, that some of the components of this space can be endowed with the structure of a smooth complex manifold.
In the following part we will apply the notions introduced in the first chapter to the derived category of coherent sheaves of a projective manifold. At the moment there is no general method for describing the structure of the space of stability conditions over any projective variety: until now there have been constructed spaces of stability conditions for surfaces and for some threefolds. We will follow in details Bridgeland’s construction of a set of stability conditions on a K3 surface and explain the main properties of this subspace. We will also sketch a construction of spaces of stability conditions on surfaces of general type. The main application of this theory is the wall-crossing method: it is a very useful tool for understanding how the stability of an object in the derived category changes when a stability condition moves continuously inside the space of stability conditions.
In the last part of the survey we will see an application of the theory to a classical problem: the Mukai’s program for curves embedded in a K3 surface. If (X,H) is a polarized K3 surface and C is a smooth curve of genus g in the linear system |H|, then the moduli space of triplets (X,H,C) is a complex manifold whose dimension is 19+g. There is a natural map to the moduli space of smooth curves of genus g which sends the triplet (X,H,C) to the curve C. It has been shown that if g=11 or g>12 this map is birational onto the surface; Mukai’s program consists in finding the rational inverse of such map. Recently Feyzbakhsh provided an elegant solution for this problem, which involved the application of wall-crossing methods. Let us fix a K3 surface (X,H) whose Picard group is generated by the ample line bundle H; Feyzbakhsh showed, using wall-crossing methods as main tool for the proof, that in this situation the moduli space of line bundle of X having fixed Mukai’s vector V is isomorphic to a Brill-Noether locus B of vector bundles on C via the map which sends a vector bundle on X to its restriction on C. By a suitable choice of V the moduli space of sheaves on X and so the Brill-Noether locus B is a K3 surface; the original surface can be reconstructed from the latter one by performing a suitable Fourier-Mukai’s transform. We concluded our thesis with a study of this problem in the case where the Picard rank of X is greater than 1; in this case the previous construction not always provides an isomorphism between the moduli space of vector bundles on X and the Brill-Noether locus on C, but often it is possible to show the injectivity of the restriction map. As a consequence we can use this as a tool for constructing curves whose Brill-Noether locus B, constructed as above, has dimension greater than its expected dimension; as an example we will provide a one-dimensional family of K3 surfaces, each of them containing the same curve C; then we may use the results previously developed in order to construct a 3-dimensional Brill-Noether locus (whose expected dimension is negative) on the curve C: using this construction we will provide a curve whose genus is 23 and whose Brill-Noether locus of rank 2 vector bundle whose degree is 44 and having at least 13 linearly independent global sections has dimension 3.