  We study inverse problems in anisotropic elasticity using tools from
algebraic geometry. The singularities of solutions to the elastic wave equation
in dimension $n$ with an anisotropic stiffness tensor have propagation
kinematics captured by so-called slowness surfaces, which are hypersurfaces in
the cotangent bundle of $\mathbb{R}^n$ that turn out to be algebraic varieties.
Leveraging the algebraic geometry of families of slowness surfaces we show
that, for tensors in a dense open subset in the space of all stiffness tensors,
a small amount of data around one polarization in an individual slowness
surface uniquely determines the entire slowness surface and its stiffness
tensor. Such partial data arises naturally from geophysical measurements or
geometrized versions of seismic inverse problems. Additionally, we explain how
the reconstruction of the stiffness tensor can be carried out effectively,
using Gr\"obner bases. Our uniqueness results fail for very symmetric (e.g.,
fully isotropic) materials, evidencing the counterintuitive claim that inverse
problems in elasticity can become more tractable with increasing asymmetry.
