--- title: Algebraic Curves description: Algebraic curves are one-dimensional algebraic varieties defined by polynomial equations. Learners will understand the geometry of projective curves, singular points, intersection theory, and the Riemann-Roch theorem, enabling them to analyze elliptic curves and Riemann surfaces. category: mathematics subcategory: algebraic-geometry difficulty: beginner, intermediate, advanced url: /subject/algebraic-curves --- # Algebraic Curves Algebraic curves are one-dimensional algebraic varieties defined by polynomial equations. Learners will understand the geometry of projective curves, singular points, intersection theory, and the Riemann-Roch theorem, enabling them to analyze elliptic curves and Riemann surfaces. ## Where to start Start with William Fulton's Algebraic Curves, released free by the author, which builds affine and projective plane curves, intersection numbers and Bezout's theorem. It is also the one to keep if you only use one resource. Afterwards, Rick Miranda's Algebraic Curves and Riemann Surfaces, a paid graduate text, approaches curves through Riemann surfaces and goes on to Serre duality and sheaf cohomology. ## Available Resources 5 Books • 1 Courses • 2 Websites ## Books ### 1. The Practice of Algebraic Curves: A Second Course in Algebraic Geometry **Author:** David Eisenbud, Joe Harris Second course on complex projective curves from two leading algebraic geometers: linear series, Riemann-Roch, curves of genus up to six, Jacobians, Hilbert schemes, moduli of curves, Severi varieties, dualizing sheaves and linkage in three-space. Fills the tier above Fulton and Miranda: how curve theory is actually used, with genus-by-genus classification and moduli. **Difficulty:** Advanced | **Language:** English | **Price:** Paid **Link:** https://www.amazon.com/dp/1470479435?tag=edmonddante07-20 **Tags:** algebraic-curves, riemann-roch, linear-series, moduli-of-curves, hilbert-schemes, jacobians ### 2. Rational Points on Elliptic Curves (2nd edition) **Author:** Joseph H. Silverman, John T. Tate Undergraduate treatment of elliptic curves as Diophantine objects: the group law, Nagell-Lutz, Mordell's theorem, curves over finite fields, integer points, complex multiplication and cryptography. The author's page carries the table of contents and 2024-updated errata. **Difficulty:** Intermediate | **Language:** English | **Price:** Paid **Link:** https://www.amazon.com/dp/3319307576?tag=edmonddante07-20 **Tags:** elliptic-curves, diophantine-equations, number-theory, mordell-theorem, arithmetic-geometry ### 3. Algebraic Curves and Riemann Surfaces **Author:** Rick Miranda Graduate text approaching curves over the complex numbers first: Riemann surfaces, meromorphic functions and divisors, then algebraic proofs of Riemann-Roch and Serre duality, closing with sheaves, Cech cohomology and invertible sheaves. Many worked examples and exercises. **Difficulty:** Advanced | **Language:** English | **Price:** Paid **Link:** https://www.amazon.com/dp/0821802682?tag=edmonddante07-20 **Tags:** riemann-surfaces, algebraic-curves, riemann-roch, sheaf-cohomology, complex-geometry ### 4. Algebraic Curves: An Introduction to Algebraic Geometry (2008 free edition) **Author:** William Fulton The canonical entry point to algebraic curves and the reference nearly every syllabus and MathOverflow reading-order answer names first. Fulton's 1969 text, revised 2008 and released free by the author. Builds affine and projective plane curves, local rings, intersection numbers, Bezout's theorem, divisors and Riemann-Roch in about 120 pages, with exercises throughout. **Difficulty:** Intermediate | **Language:** English | **Price:** Free **Link:** https://people.reed.edu/~davidp/332/CurveBook.pdf **Tags:** algebraic-curves, plane-curves, bezout-theorem, intersection-multiplicity, riemann-roch, projective-geometry ### 5. Algebraic Curves **Author:** William Fulton William Fulton's concise graduate-level introduction to algebraic geometry through plane curves: affine and projective varieties, local rings and intersection multiplicities, Bézout's theorem, Max Noether's theorem, and Riemann-Roch. Assumes abstract algebra; prepares readers for Hartshorne-level texts. **Difficulty:** Advanced | **Language:** English | **Price:** Paid **Link:** https://www.amazon.com/dp/0805330828?tag=edmonddante07-20 **Tags:** algebraic-geometry, algebraic-curves, projective-geometry, intersection-theory, riemann-roch ## Courses ### 1. Plane Algebraic Curves — Class Notes **Author:** Andreas Gathmann Course notes from RPTU Kaiserslautern: affine and projective plane curves, intersection multiplicities, Bezout's theorem, functions and divisors, elliptic curves and the Riemann-Roch theorem across 71 pages, with exercises and lecture recordings. **Difficulty:** Intermediate | **Language:** English | **Price:** Free **Link:** https://agag-gathmann.math.rptu.de/pac **Tags:** plane-curves, bezout-theorem, projective-geometry, elliptic-curves, riemann-roch ## Websites ### 1. 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 ### 2. nLab Collaborative wiki for mathematics, physics and philosophy written from a category-theoretic viewpoint, with cross-linked entries on categories, functors, adjunctions, topos theory, higher category theory and homotopy theory. Readers can look up precise definitions, examples and references for advanced structural mathematics. **Difficulty:** Advanced | **Language:** English | **Price:** Free **Link:** https://ncatlab.org **Tags:** category-theory, higher-category-theory, topos-theory, homotopy-theory, reference-wiki --- *This content is part of Dantes.io - Your Treasure Map to Knowledge* *Curated by humans at Dantes.io. Personal study use welcome; republishing this curation requires permission (team@dantes.io).* View this page online: https://dantes.io/subject/algebraic-curves