Resolutions of linear $p$-cyclic quotient singularities

Let $k$ be an algebraically closed field of positive characteristic $p$, and let $C_p$ act linearly on a finite-dimensional $k$-vector space $V$, with indecomposable Jordan summands $V_{d_i}$ of dimension $d_i$. We study projective (crepant) resolutions of $V\mathrm{/}{C}_p$ by constructing a model using the invariant weighted blowup. For $V=V_2^{\oplus n}$, our model is smooth and is the normalization of a blowup of $V\mathrm{/}{C}_p$. It has a unique exceptional divisor of discrepancy $n-p$, such that our model is a crepant resolution when $n=p$. In almost all other cases when the quotient is canonical but not terminal, the invariant weighted blowup model does not produce crepant resolutions, but gives the unique nontrivial projective crepant birational model of the quotient. Consequently, up to trivial summands, we classify the $p$-cyclic linear quotients admitting a projective crepant resolution.