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If is a smooth projective curve of genus at least three, then the smooth projective surface contains no curve of geometric genus below three. On the normalization of any curve in the product, at least one coordinate projection to is nonconstant, and the Riemann-Hurwitz formula cannot decrease genus.
For a finite morphism of smooth curves, the ramification divisor isThe Riemann-Hurwitz formula states .
A plane curve is the zero set of a polynomial in two variables, or a one-dimensional curve embedded in a plane.
A plane cubic is nonsingular when its homogeneous equation and all first partial derivatives have no common projective zero.
The gonality of a smooth projective curve is the least degree of a nonconstant morphism from the curve to the projective line.
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.
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
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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.
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Figure 3. Visual Studio Code extension installation.
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