Instanton sheaves of low charge on Fano threefolds

Abstract: Let $X$ be a Fano threefold of Picard number one and of index $2+h, \ h=0,1$. An \textit{instanton sheaf of charge $k$ on $X$} is defined as a semi-stable rank 2 torsion free sheaf $F$ with Chern classes $c_1=-h, \ c_2=k, \ c_3=0$ and such that $F(-1)$ has no cohomology. Locally free instantons, originally defined on the projective space and later generalised on other Fano threefolds $X$, had been largely studied from several authors in the past years; their moduli spaces present an extremely rich geometry and useful applications to the study of curves on $X$. In this talk I will illustrate several features of non-locally free instantons of low charge on 3 dimensional quadrics and cubics. I will focus in particular on the role that they play in the study of the Gieseker-Maruyama moduli space $M_X(2;-h,k,0)$ and describe how we can still relate these sheaves to curves on $X$.

algebraic geometry

Audience: researchers in the topic

Brazilian algebraic geometry seminar

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