  We study the problem of characterizing linear preserver subgroups of
algebraic varieties, with a particular emphasis on secant varieties and other
varieties of tensors. We introduce a number of techniques built on different
geometric properties of the varieties of interest. Our main result is a simple
characterization of the linear preservers of secant varieties of Segre
varieties in many cases, including $\sigma_r((\mathbb{P}^{n-1})^{\times k})$
for all $r \leq n^{\lfloor k/2 \rfloor}$. We also characterize the linear
preservers of several other sets of tensors, including subspace varieties, the
variety of slice rank one tensors, symmetric tensors of bounded Waring rank,
the variety of biseparable tensors, and hyperdeterminantal surfaces.
Computational techniques and applications in quantum information theory are
discussed. We provide geometric proofs for several previously known results on
linear preservers.
