---
title: Algebraic Varieties
description: Algebraic varieties are geometric objects defined as the zero loci of sets of polynomial equations. Learners will understand the Zariski topology, coordinate rings, dimension theory, and morphisms, enabling them to study geometric properties using algebraic techniques.
category: mathematics
subcategory: algebraic-geometry
difficulty: beginner, intermediate, advanced
url: /subject/algebraic-varieties
---

# Algebraic Varieties

Algebraic varieties are geometric objects defined as the zero loci of sets of polynomial equations. Learners will understand the Zariski topology, coordinate rings, dimension theory, and morphisms, enabling them to study geometric properties using algebraic techniques.

## Available Resources

3 Books • 1 Courses • 3 Websites

## Websites

### 1. Algebraic Geometry Notes by Johan de Jong

Lecture notes on varieties

**Difficulty:** Beginner | **Price:** Free

**Link:** https://www.math.columbia.edu/~dejong/heart.html

### 2. Stacks Project

**Author:** The Stacks Project Authors

A collaborative, open-source reference on algebraic geometry run from Columbia University, with thousands of pages building from commutative algebra to schemes and algebraic stacks. Every result has a stable tag, so readers can look up precise statements and complete proofs.

**Difficulty:** Advanced | **Price:** Free

**Link:** https://stacks.math.columbia.edu/

**Tags:** algebraic-geometry, schemes, commutative-algebra, algebraic-stacks

### 3. Wolfram MathWorld

**Author:** Eric W. Weisstein

MathWorld is an online mathematics encyclopedia from Wolfram Research offering detailed, browsable articles on topics across the math spectrum, including algebra, geometry, calculus, and number theory. Each entry includes definitions, theorems, formulas, diagrams, worked examples, and links to further reading.

**Difficulty:** Intermediate | **Language:** English | **Price:** Free

**Link:** https://mathworld.wolfram.com

**Tags:** mathematics-reference, encyclopedia, abstract-algebra, number-theory, geometry

## Courses

### 1. Math 216: Foundations of Algebraic Geometry (The Rising Sea)

**Author:** Ravi Vakil

Ravi Vakil's Stanford course site hosting free drafts of The Rising Sea, a graduate text on Grothendieck-style algebraic geometry: sheaves, schemes, morphisms, dimension, smoothness, quasicoherent sheaves and cohomology, with extensive exercises. Readers learn to work fluently with schemes and the modern foundations of the field.

**Difficulty:** Advanced | **Price:** Free

**Link:** https://math216.wordpress.com/

**Tags:** algebraic-geometry, schemes, sheaves, cohomology, commutative-algebra

## Books

### 1. Algebraic Geometry

**Author:** Robin Hartshorne

Standard graduate textbook (Springer GTM 52) that develops varieties, then schemes and sheaf cohomology, and applies them to curves and surfaces. After working the exercises, readers can use scheme-theoretic language fluently and compute cohomology, divisors and Riemann-Roch on curves.

**Difficulty:** Intermediate | **Language:** English | **Price:** Paid

**Link:** https://www.amazon.com/dp/1441928073?tag=edmonddante07-20

**Tags:** books, mathematics-statistics, algebraic-geometry

### 2. Basic Algebraic Geometry 1

**Author:** Igor R. Shafarevich

First volume of Shafarevich's classic text, covering varieties in projective space: regular and rational maps, dimension, local properties and nonsingularity, divisors and differential forms, and algebraic curves and surfaces. Readers gain geometric intuition for varieties before tackling schemes in heavier texts.

**Difficulty:** Intermediate | **Language:** English | **Price:** Paid

**Link:** https://www.amazon.com/dp/3540548122?tag=edmonddante07-20

**Tags:** books, mathematics-statistics, algebraic-geometry

### 3. Introduction to Algebraic Geometry

**Author:** Brendan Hassett

Undergraduate-level introduction built around computation: Gröbner bases, affine and projective varieties, elimination, the Nullstellensatz, irreducible decomposition and rational maps. Readers learn to compute with polynomial ideals and connect algebraic operations to the geometry of the varieties they define.

**Difficulty:** Intermediate | **Language:** English | **Price:** Paid

**Link:** https://www.amazon.com/dp/0521691419?tag=edmonddante07-20

**Tags:** books, mathematics-statistics, algebraic-geometry

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