A perfect obstruction theory for moduli of coherent systems

Let $C$ be a curve of genus $g$. A coherent system on $C$ is a pair $(E,V)$, where $E$ is a finite rank vector bundle on $C$ and $V$ is a linear subspace of the space of global sections of $E$. The type of a coherent system $(E,V)$ is a triple $(n,d,k)$, where $n$ is the rank of $E$, $d$ is the degree of $E$ and $k$ is the dimension of $V$. The notion of stability for a coherent system $(E,V)$ differs from the stability of the bundle $E$ and depends on the choice of a real parameter $\alpha$. The moduli space of $\alpha$-stable coherent systems of type $(n,d,k)$ has an expected dimension $\beta = \beta(n,d,k)$ which depends on the genus of the curve $C$ and on the type of the coherent systems. We construct a perfect obstruction theory for the moduli spaces of $\alpha$-stable coherent systems which has rank equal to the expected dimension $\beta$. In our construction we do not fix one curve, but we work on families of Gorenstein projective curves.