---
tag: Z5G47MQ99B
kind: concept
name: Projective morphism
msc: [14A15]
specializes: [9537RKBZPP]
sources:
  - url: https://en.wikipedia.org/wiki/Glossary_of_algebraic_geometry
---

## Summary

A projective morphism $X \to Y$ is a morphism that embeds $X$ as a closed subscheme of a projective space over $Y$.
It is the relative form of a projective variety: a family of projective schemes parametrized by $Y$.
Projective morphisms are proper, and most proper morphisms met in practice are projective.

## Explanations

### L1 human

A projective variety is cut out by homogeneous polynomial equations inside projective space.
A projective morphism to $Y$ is the same thing with coefficients that vary over $Y$.
For example, the equations $y^2 z = x^3 + t x z^2 + z^3$ in $\mathbb{P}^2$, with a parameter $t$, describe a family of plane cubic curves over the $t$-line.
The map that sends each point of the family to its value of $t$ is projective.