Is it correct to say $ x^3+2x+1=y^2 $ is an elliptic curve?

Question

I'm a bit confused about the definition on elliptic curve. For example, can we say that $x^3+2x+1=y^2$ is an elliptic curve? My opinion is that it is not an elliptic curve as the definition given in the Algebraic Geometry and Arithmetic Curves requires a privileged rational point and it has to be isomorphic to a closed subvariety of $\mathbb{P}_k^2$. But what is the experts opinion about calling a curve an elliptic curve if privileged rational point or closed subvariety is not mentioned?

Answer 1

It's a standard convention in the field to write $f(x, y) = 0$, which a priori defines an affine curve sitting in $\mathbb{A}^2$, with no distinguished points, when one really means the projective closure of this curve sitting in $\mathbb{P}^2$, with distinguished points the points at infinity. If $f(x, y) = 0$ has the form

$$y^2 = x^3 + ax + b$$

where the polynomial on the RHS has nonzero discriminant, then this projective closure is a smooth projective curve of genus $1$, and moreover there is exactly one point at infinity. This is the distinguished point.

(Using subvarieties of $\mathbb{P}^2$ is a crutch. In the more general context of algebraic curves one should talk abstractly about smooth projective curves without a choice of embedding.)