A projective variety is a type of geometric object studied in algebraic geometry. It is defined as the set of solutions to a system of polynomial equations in projective space, which is a higher-dimensional space that allows for the representation of points at infinity. Projective varieties can be thought of as the "shapes" formed by these solutions, and they are important for understanding the properties of algebraic equations. These varieties can be classified into different types, such as projective curves, projective surfaces, and higher-dimensional varieties. They are often studied using tools from commutative algebra and homological algebra, which help to analyze their structure and relationships. Projective varieties play a crucial role in various areas of mathematics, including number theory and mathematical physics.