  We study algebraic subvarieties of strata of differentials in genus zero
satisfying algebraic relations among periods. The main results are Ax-Schanuel
and Andr\'e-Oort-type theorems in genus zero. As a consequence, one obtains
several equivalent characterizations of bi-algebraic varieties. It follows that
bi-algebraic varieties in genus zero are foliated by affine-linear varieties.
Furthermore, bi-algebraic varieties with constant residues in strata with only
simple poles are affine-linear. Additionally, we produce infinitely many new
linear varieties in strata of genus zero, including infinitely many new
examples of meromorphic Teichm\"uller curves.
