Intersection numbers

Intersection numbers are a way mathematicians quantify how different geometric shapes, like curves or surfaces, intersect within a space. Think of them as counting the points where the shapes cross or touch, but in a more advanced setting that considers multiplicities and directions. For example, if two curves cross at a point, the intersection number might be 1 if they just meet once neatly, or higher if they touch more tangentially or intersect multiple times in a more complex way. These numbers help understand the geometric and algebraic relationships between shapes in spaces such as algebraic varieties.