  We introduce the problem of constructing explicit variety evasive subspace
families. Given a family $\mathcal{F}$ of subvarieties of a projective or
affine space, a collection $\mathcal{H}$ of projective or affine $k$-subspaces
is $(\mathcal{F},\epsilon)$-evasive if for every $\mathcal{V}\in\mathcal{F}$,
all but at most $\epsilon$-fraction of $W\in\mathcal{H}$ intersect every
irreducible component of $\mathcal{V}$ with (at most) the expected dimension.
The problem of constructing such an explicit subspace family generalizes both
deterministic black-box polynomial identity testing (PIT) and the problem of
constructing explicit (weak) lossless rank condensers.
  Using Chow forms, we construct explicit $k$-subspace families of polynomial
size that are evasive for all varieties of bounded degree in a projective or
affine $n$-space. As one application, we obtain a complete derandomization of
Noether's normalization lemma for varieties of low degree in a projective or
affine $n$-space. In another application, we obtain a simple polynomial-time
black-box PIT algorithm for depth-4 arithmetic circuits with bounded top fan-in
and bottom fan-in that are not in the Sylvester-Gallai configuration, improving
and simplifying a result of Gupta (ECCC TR 14-130).
  As a complement of our explicit construction, we prove a tight lower bound
for the size of $k$-subspace families that are evasive for degree-$d$ varieties
in a projective $n$-space. When $n-k=n^{\Omega(1)}$, the lower bound is
superpolynomial unless $d$ is bounded. The proof uses a dimension-counting
argument on Chow varieties that parametrize projective subvarieties.
