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A degree- projective hypersurface is the closed subscheme cut out by one nonzero homogeneous polynomial of degree .
The zero set of a nonconstant polynomial in affine variables has dimension . Passing to the square-free part gives its radical principal ideal.
The Krull dimension of an affine variety is the supremum of the lengths of strict chains of irreducible closed subsets, equivalently the Krull dimension of its coordinate ring. For an irreducible affine variety it equals the minimum Zariski tangent-space dimension.
For , the Zariski tangent space is the common kernel at of the differentials of all polynomials in .
Genus distinguishes equal-degree projective curves by 
Codex 0 Created 2026-09-24 Updated 2026-09-24
Codex 0 Created 2026-09-24 Updated 2026-09-24A twisted cubic and a smooth plane cubic embedded in both have degree three, but their genera are zero and one. Thus equal projective degree does not imply isomorphism.
The twisted cubic is the image ofin . Its ideal is generated by the three minors ofand it has degree three.
If the Hilbert polynomial of the homogeneous coordinate ring of a projective curve isthen . Equivalently, a sufficiently general hyperplane section has scheme-theoretic length .
For an algebraically closed field and an ideal , the strong Hilbert Nullstellensatz isIts weak form says that an ideal with empty affine zero set is the unit ideal. Equivalently, if finitely many polynomials have no common zero, a polynomial combination of them equals one.
Every finitely generated algebra over a field has algebraically independent elements such that is finite as a module over the polynomial subalgebra . The integer is the Krull dimension of .
A Zariski-open set is the complement of a Zariski-closed set. A distinguished affine open has the form .
A Zariski-closed set is a common zero set of polynomials. Arbitrary intersections and finite unions of such sets are again Zariski closed.
A nonempty topological space is irreducible when it is not the union of two proper closed subsets. Equivalently, every two nonempty open subsets intersect.
An algebraic variety is a geometric space locally described by polynomial equations, together with its regular functions.
A Cartier divisor is locally represented by nonzero rational functions whose ratios on overlaps are regular units. It is principal when one global rational function represents all local data.
A Weil divisor on a Noetherian integral scheme is a finite integer combination of integral codimension-one closed subschemes.
A ringed space is a topological space equipped with a sheaf of rings . A morphism consists of a continuous map and a compatible morphism .
An associative algebra is an algebra over a field whose multiplication satisfies . Unless stated otherwise, the algebras considered here have a multiplicative identity.
A Hopf algebra is a compatible algebra and coalgebra equipped with a counit and antipode. Coordinate rings of affine algebraic groups are commutative Hopf algebras.
For a vector space with quadratic form , its Clifford algebra is generated by vectors subject to . In a basis this is equivalently .
Pinned article: Introduction to the OurBigBook Project
Welcome to the OurBigBook Project! Our goal is to create the perfect publishing platform for STEM subjects, and get university-level students to write the best free STEM tutorials ever.
Everyone is welcome to create an account and play with the site: ourbigbook.com/go/register. We belive that students themselves can write amazing tutorials, but teachers are welcome too. You can write about anything you want, it doesn't have to be STEM or even educational. Silly test content is very welcome and you won't be penalized in any way. Just keep it legal!
Intro to OurBigBook
. Source. We have two killer features:
- topics: topics group articles by different users with the same title, e.g. here is the topic for the "Fundamental Theorem of Calculus" ourbigbook.com/go/topic/fundamental-theorem-of-calculusArticles of different users are sorted by upvote within each article page. This feature is a bit like:
- a Wikipedia where each user can have their own version of each article
- a Q&A website like Stack Overflow, where multiple people can give their views on a given topic, and the best ones are sorted by upvote. Except you don't need to wait for someone to ask first, and any topic goes, no matter how narrow or broad
This feature makes it possible for readers to find better explanations of any topic created by other writers. And it allows writers to create an explanation in a place that readers might actually find it.
Figure 1. Screenshot of the "Derivative" topic page. View it live at: ourbigbook.com/go/topic/derivativeVideo 2. OurBigBook Web topics demo. Source. - local editing: you can store all your personal knowledge base content locally in a plaintext markup format that can be edited locally and published either:This way you can be sure that even if OurBigBook.com were to go down one day (which we have no plans to do as it is quite cheap to host!), your content will still be perfectly readable as a static site.
- to OurBigBook.com to get awesome multi-user features like topics and likes
- as HTML files to a static website, which you can host yourself for free on many external providers like GitHub Pages, and remain in full control
Figure 2. You can publish local OurBigBook lightweight markup files to either OurBigBook.com or as a static website.
Figure 3. Visual Studio Code extension installation.
Figure 5. . You can also edit articles on the Web editor without installing anything locally. Video 3. Edit locally and publish demo. Source. This shows editing OurBigBook Markup and publishing it using the Visual Studio Code extension. 
- Infinitely deep tables of contents:
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