Hormander showed that there is a generic set of scalar linear PDE's that can be studied using general techniques, known as microlocal analysis. This can be linked to algebraic geometry as follows: Any scalar linear partial differential operator of order $k$ on an open set in $\mathbb{R}n$ can be written as
$$
Pu = \sum_{|\alpha|\le k} a^\alpha\partial_\alpha u,
$$
where each coefficient $a^\alpha$ is a smooth function, $\alpha = (\alpha_1, \dots, \alpha_n)$ and $\partial_\alpha = (\partial_1)^{\alpha_1}\cdots(\partial_n)^{\alpha_n}$. If this is studied using the Fourier transform, then a natural object to study turns out to the principal symbol
$$
\sigma(x,\xi) = \sum_{|\alpha| = k} a^\alpha(x)\xi_\alpha,
$$
where $\xi = (\xi_1, \dots, \xi_n) \in \mathbb{R}^n$ and $\xi_\alpha = (\xi_1)^{\alpha_1}\cdots(\xi_n)^{\alpha_n}$. For each $x$, this is a homogeneous polynomial of degree $k$ and therefore its zero set is a real algebraic variety on $\mathbb{R}P^{n-1}$. This is known as the characteristic variety. Hormander proved, if the characteristic variety is generic in a suitable sense, regularity estimates, local existence of solutions, and many other things about solutions to equations defined using such operators. However, PDEs most studied have symbols lying in a subvariety of very high codimension, and the techniques used by Hormander are used outside the field of microlocal analysis in only a few specialized areas (e.g., scattering theory, inverse problems). The PDEs with most impact are elliptic, hyperbolic, and parabolic PDEs. Elliptic and most hyperbolic PDEs are generic in Hormander's sense, but parabolic PDEs are not.
