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 generalization of algebraic curves and surfaces, and they play a crucial role in understanding the properties of algebraic equations. These varieties are often studied using tools from both algebra and geometry, such as homogeneous coordinates and intersection theory. They can exhibit interesting features, such as singularities and different dimensions, and are classified into various types, including projective spaces and Grassmannians. Understanding projective varieties helps mathematicians explore deeper connections between algebraic equations and geometric shapes.