Bertini's Theorem

Bertini's Theorem states that if you have a smooth, well-behaved algebraic shape (called a variety), then most of the ways you cut it with a simple geometric shape (like a line or a plane) will produce a new shape that is also smooth and without singularities. In other words, for a generic choice of a low-degree shape intersecting a smooth variety, the intersection will itself be smooth and nicely structured. This theorem helps mathematicians understand how complex shapes behave when sliced or intersected with simpler figures, ensuring that "most" intersections preserve good geometric properties.