Starts 27 Apr 2026 14:30
Ends 27 Apr 2026 16:00
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Basic Notions Seminar - Part I: Moduli spaces of sheaves on threefolds. Part II: Moduli spaces of rank 2 torsion-free sheaves with quasi-maximal third Chern class
ICTP
Leonardo Building - Euler Lecture Hall
ABSTRACT PART I:
While moduli spaces of sheaves on curves and surfaces are relatively well understood, understanding the case of varieties of dimension 3 is considered a hard problem. After giving a quick introduction to moduli spaces and sheaves, I will present the main problems and the challenges involved, as well as some of the results we have proved recently.
While moduli spaces of sheaves on curves and surfaces are relatively well understood, understanding the case of varieties of dimension 3 is considered a hard problem. After giving a quick introduction to moduli spaces and sheaves, I will present the main problems and the challenges involved, as well as some of the results we have proved recently.
ABSTRACT PART II:
Hartshorne established the irreducibility and smoothness of the Gieseker-Maruyama moduli spaces parameterizing semistable rank 2 reflexive sheaves on P3 with c_1=-1 and the maximal third Chern class c_3 = c_2^2. Later, Okonek and Spindler and Schmidt, in an independent work, prove that this is still true for torsion-free sheaves. However, a gap theorem by Miró-Roig shows that the moduli space of reflexive sheaves is empty for a certain range on the c_3, including the quasi-maximal case c_3 = c_2^2 - 2. Despite the absence of reflexive sheaves, torsion-free sheaves do exist in this range, Almeida, Jardim, and Tikhomirov constructed a family of components for these spaces. In this talk we will use the modular Hartshorne-Serre correspondence to prove the irreducibility of M(-1, c_2, c_2^2 - 2) for c_2 =3.
Hartshorne established the irreducibility and smoothness of the Gieseker-Maruyama moduli spaces parameterizing semistable rank 2 reflexive sheaves on P3 with c_1=-1 and the maximal third Chern class c_3 = c_2^2. Later, Okonek and Spindler and Schmidt, in an independent work, prove that this is still true for torsion-free sheaves. However, a gap theorem by Miró-Roig shows that the moduli space of reflexive sheaves is empty for a certain range on the c_3, including the quasi-maximal case c_3 = c_2^2 - 2. Despite the absence of reflexive sheaves, torsion-free sheaves do exist in this range, Almeida, Jardim, and Tikhomirov constructed a family of components for these spaces. In this talk we will use the modular Hartshorne-Serre correspondence to prove the irreducibility of M(-1, c_2, c_2^2 - 2) for c_2 =3.