Given a finite \(p\)-group \(G\) acting on a smooth projective curve \(X\) over an algebraically closed field \(k\) of characteristic \(p\), the dimension of the tangent space of the associated equivariant deformation functor is equal to the dimension of the space of coinvariants of \(G\) acting on the space \(V\) of global holomorphic quadratic differentials on \(X\). We apply known results about the Galois module structure of Riemann-Roch spaces to compute this dimension when \(G\) is cyclic or when the action of \(G\) on \(X\) is weakly ramified. Moreover we determine certain subrepresentations of \(V\), called \(p\)-rank representations.
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