projective scheme
a scheme 𝑋→𝑆 over some base scheme 𝑆 that factors as a closed immersion 𝑋↪ℙⁿ_𝑆→𝑆
projective variety
algebraic variety defined within a projective space
a scheme 𝑋→𝑆 over some base scheme 𝑆 that factors as a closed immersion 𝑋↪ℙⁿ_𝑆→𝑆