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A plane cubic is nonsingular when its homogeneous equation and all first partial derivatives have no common projective zero.

A rational map is a morphism on a dense open subset, considered up to agreement on a smaller dense open subset.

It is the projective-source-and-target case of a rational map of algebraic varieties.

A hyperelliptic curve of genus at least two admits a degree-two map to ${\mathbb{P}}^1$ and can be represented in characteristic other than two by an equation $y^2=f(x)$ with $f$ square-free.

The gonality of a smooth projective curve is the least degree of a nonconstant morphism from the curve to the projective line.

A divisor is a finite formal integer combination of closed points. For a divisor $D$,

This is the one-dimensional case of normalization of an algebraic variety, constructed from integral closures of coordinate rings.

The normalization of an irreducible algebraic curve is a normal, hence regular, curve with a finite birational morphism to the original curve. Over a perfect field this regular curve is smooth.

Irreducible varieties are birational exactly when they have isomorphic function fields.

This equivalence is the relation studied by birational geometry.

A determinantal variety is cut out by minors imposing an upper bound on matrix rank.

A degree-$d$ projective hypersurface $X_d\subseteq {\mathbb{P}}_k^n$ is the closed subscheme cut out by one nonzero homogeneous polynomial of degree $d$.

This is a homogeneous-polynomial hypersurface in projective space.

The zero set of a nonconstant polynomial in $n$ affine variables has dimension $n-1$. Passing to the square-free part gives its radical principal ideal.

This is a polynomial-defined hypersurface in affine space.

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 $P\in X\subseteq {\mathbb{A}}^n$, the Zariski tangent space is the common kernel at $P$ of the differentials of all polynomials in $I(X)$.

A twisted cubic and a smooth plane cubic embedded in ${\mathbb{P}}^3$ 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 ${\mathbb{P}}^3$. Its ideal is generated by the three $2\times 2$ minors ofand it has degree three.

If the Hilbert polynomial of the homogeneous coordinate ring of a projective curve isthen $\mathrm{deg}C=d$. Equivalently, a sufficiently general hyperplane section has scheme-theoretic length $d$.

The graph $\{(x,e^x):x\in \mathbb{C}\}$ is Zariski dense in ${\mathbb{A}}_{\mathbb{C}}^2$. If ${\sum }_j{p}_j(x)e^{jx}=0$, repeated application of $D-m$ removes the largest exponential term while acting injectively on the others, proving inductively that every $p_j$ vanishes.

For an algebraically closed field $k$ and an ideal $I\subseteq k[x_1,\dots ,x_n]$, 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 $A$ over a field has algebraically independent elements $y_1,\dots ,y_d$ such that $A$ is finite as a module over the polynomial subalgebra $k[y_1,\dots ,y_d]$. The integer $d$ is the Krull dimension of $A$.

A Zariski-open set is the complement of a Zariski-closed set. A distinguished affine open has the form $D(f)=\{x:f(x)\ne 0\}$.

A Zariski-closed set is a common zero set of polynomials. Arbitrary intersections and finite unions of such sets are again Zariski closed.