Characteristic of a field and algebraic curves

Question

In what way the characteristic of a field influences the curves defined over that field? For example, let $C$ be the curve defined by $X^3+Y^3+Z^3=0$ over an algebraically closed field $K$. What happen to the curve if $char(K)=3$ or $char(K)\ne 3$? Thank you!

Answer 1

Over characteristic $p=3$, $X^3+Y^3+Z^3 = (X+Y+Z)^3$.