Before algebraic geometry was developed sufficiently and the Riemann-Roch was proved with its present power, the Lefschetz principle was used to dispose of many statements in algebraic geometry.
For instance: Define an elliptic curve over a field as a curve in the Weierstrass form with nonzero determinant. Upon this define the addition and inverse laws using the chord-and-tangent process, obtaining algebraic expressions. To show that the elliptic curve is a group, you have to show the addition is associative. One way is a very tedious verification of the identities.
Another way is to use elliptic functions to prove the identity in the complex case. Since the algebraic group law holds true over the complex numbers, it is satisfied by an infinite number of algebraically independent solutions, and therefore the group law must be true in universality, over any field whatsoever. Of course this needs to be made precise with Lefschetz principle.
But later algebraic geometry developed and it was possible to prove statements without relying on the Lefschetz principle. For instance, the group law on elliptic curve is always a consequence of the Riemann-Roch.