Birational geometry

Birational geometry is a branch of algebraic geometry that studies the relationships between algebraic varieties, which are geometric shapes defined by polynomial equations. Essentially, it focuses on understanding when two varieties can be considered equivalent, despite possibly being different in a technical sense. This is done using birational maps, which are like functions that relate these shapes, allowing us to transform and compare them. The field explores properties that remain intact under these relationships, aiming to classify varieties and understand their geometric and arithmetic features, which has implications in both mathematics and theoretical physics.