General curves always have finitely many points with rational coordinates?

Question

In the latest episode of Numberphile, Professor Várilly-Alvarado made the following claim

General curves always have finitely-many points with rational coordinates

This is obviously false for general curves - take, for example, any curve which contains any non-zero-width segment of the number line. The professor himself also showed earlier that it's not true for circles and other "rational curves".

So, what is the actual claim here? What is the definition of "general curve"? Why are they called "general" when they seem to be a special case?

Answer 1

Here, a curve is an algebraic curve, i.e. an irreducible curve that is the zero locus of a collection of polynomials. Algebraic curves have an invariant called the genus, which is a non-negative integer, and for a curve which is defined over $\Bbb Q$ (i.e. the zero locus of polynomial equations with rational coefficients) the genus of the curve tells you how many rational points it can have:

• Curves of genus 0 (rational curves) have either 0 rational points or infinitely many rational points. (If any irreducible algebraic curve contains a line segment, it is actually a line, so this is where the curves containing "non-zero-width segment of the number line" fit.)
• Curves of genus 1 may have 0 rational points, or if they have a rational point they are an elliptic curve and have a finitely-generated abelian group of rational points by the Mordell-Weil theorem.
• Curves of genus $\geq 2$ are curves of general type and have only finitely many rational points by Falting's theorem.