Editing Scheme (mathematics) (section)

=== Arithmetic surfaces ===
If we consider a polynomial <math>f \in \mathbb{Z}[x,y]</math> then the affine scheme <math>X = \operatorname{Spec}(\mathbb{Z}[x,y]/(f))</math> has a canonical morphism to <math>\operatorname{Spec}\mathbb{Z}</math> and is called an [[arithmetic surface]]. The fibers <math>X_p = X \times_{\operatorname{Spec}(\mathbb{Z})}\operatorname{Spec}(\mathbb{F}_p)</math> are then algebraic curves over the finite fields <math>\mathbb{F}_p</math>. If <math>f(x,y) = y^2 - x^3 + ax^2 + bx + c</math> is an [[elliptic curve]], then the fibers over its discriminant locus, where  <math display="block">\Delta_f = -4a^3c + a^2b^2 + 18abc - 4b^3 - 27c^2 = 0 \ \text{mod}\ p,</math>are all singular schemes.<ref>{{Cite web |title=Elliptic curves |url=https://homepages.warwick.ac.uk/~maskal/MA426_EllipticCurves_2018.pdf |page=20}}</ref> For example, if <math>p</math> is a prime number and <math display="block">X = \operatorname{Spec}
    \frac{\mathbb{Z}[x,y]}{(y^2 - x^3 - p)}</math> then its discriminant is <math>-27p^2</math>. This curve is singular over the prime numbers <math>3, p</math>.