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Best resources to learnAlgebraic Curves

Study sheet

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.

8 resources · · How we curate

🎓Courses(1)

course

Plane Algebraic Curves — Class Notes

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.

intermediate🇬🇧

📚Books(5)

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bookPaid

The Practice of Algebraic Curves: A Second Course in Algebraic Geometry

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.

advanced🇬🇧
bookPaid

Rational Points on Elliptic Curves (2nd edition)

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.

intermediate🇬🇧
bookPaid

Algebraic Curves and Riemann Surfaces

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.

advanced🇬🇧
book

Algebraic Curves: An Introduction to Algebraic Geometry (2008 free edition)

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.

intermediate🇬🇧
bookPaid

Algebraic Curves

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.

advanced🇬🇧

🌐Websites(2)

website

Wolfram MathWorld

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.

intermediate🇬🇧
website

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.

advanced🇬🇧
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Frequently Asked Questions

Where should I start learning Algebraic Curves?

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.

What are the best free resources to learn Algebraic Curves?

Dantes has curated 8 resources for Algebraic Curves, including 5 books, 1 courses, 2 websites. All resources are hand-picked for quality — no algorithmic filler. Browse the full list above to find the format that works best for you.

Is Algebraic Curves hard to learn?

Algebraic Curves is approachable at the beginner level — there are resources here specifically for those starting from scratch. As you progress, intermediate and advanced material is also available to take your skills further.

What types of Algebraic Curves learning resources are available on Dantes?

For Algebraic Curves, Dantes has curated 5 books, 1 courses, 2 websites. Each resource type serves a different learning style: videos and YouTube for visual learners, books for depth, courses for structured progression, and websites for quick reference.

How does Dantes select Algebraic Curves resources?

Dantes is an algorithm-free learning directory. Resources are hand-curated based on quality, accuracy, and usefulness — not engagement metrics or paid placements. The goal is to surface the best learning material for Algebraic Curves, whether it's a free YouTube series, a classic textbook, or an open courseware from a top university.

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