Castelnuovo's theorem

Castelnuovo's theorem pertains to algebraic geometry and states that any algebraic curve of degree four in a projective plane (a certain mathematical setting) is uniquely determined by six points lying on it, provided these points are in general position (not all on a simple conic). Essentially, it highlights the special structure and properties of quartic curves, demonstrating that their shape and nature can be fully understood through a specific set of points, reflecting how geometric objects of degree four are intricately linked to the configurations of points they contain.