Elliptic curve
If we consider a polynomial formula_5 then the affine scheme formula_6 has a canonical morphism to formula_7 and is called an Arithmetic surface. The fibers formula_8 are then algebraic curves over the finite fields formula_9. If formula_10 is an Elliptic curve then the fibers over its discriminant locus generated by formula_11 where formula_12 are all singular schemes. For example, if formula_13 is a prime number and formula_14 then its discriminant is formula_15. In particular, this curve is singular over the prime numbers formula_16.