On a family of special linear systems on algebraic curves

$\mathcal{W}_6^2$ is a scheme parametrizing pairs $L \to C$ of smooth algebraic curves $C$ of genus 10 together with line bundles $L$ of degree 6 such that ${H^0}\left( {C,L} \right) \geqslant 3$. It is shown that one of the irreducible components of this scheme is nonreduced at every point.