A projective variety is a fundamental concept in algebraic geometry, representing a type of geometric object defined by polynomial equations. It is formed within a projective space, which is a mathematical structure that extends the idea of ordinary space by adding "points at infinity." This allows for a more comprehensive understanding of geometric properties, particularly for curves and surfaces. Projective varieties can be studied using tools from both algebra and geometry, making them essential for exploring relationships between different mathematical objects. They are closely related to concepts like affine varieties and homogeneous coordinates, which help in analyzing their structure and behavior in various dimensions.