Bézout's theorem

Two projective plane curves of degrees $m$ and $n$ with no common irreducible component have total intersection multiplicity $mn$. In particular, two nonempty projective plane curves always intersect.

For projective plane curves without a common irreducible component, project from a point outside both curves. In coordinates where that point is $[1:0:0]$, their defining homogeneous polynomials have constant nonzero leading coefficients in $X$. The resultant in $X$ is nonzero by Gauss's lemma, and is homogeneous of degree $\mathrm{deg}C\mathrm{deg}D$ in the remaining two coordinates. Every intersection projects to one of its zeros. Each projection line has only finitely many intersections with the curves, so their intersection is finite. Choose the center again outside all joining lines of pairs of intersection points; the projection is now injective on that finite set, proving the bound.

A projective plane curve is a projective curve embedded in the projective plane, usually given by one homogeneous equation.

A degree-two projective plane curve, allowing singular and nonreduced cases. In characteristic different from two, write its equation as $z^{\mathsf{T}}Mz$ with $M$ a symmetric matrix. It is smooth exactly when $M$ has matrix rank three. Over an algebraically closed field, rank two gives two distinct projective lines, while rank one gives a double line.

A degree-four projective plane curve. Its arithmetic genus is three by the genus-degree formula; singularities can lower the genus of its normalization of an algebraic curve.

Translate the point to the origin of an affine chart and expand a local defining polynomial as $\sum a_{ij}{x}^i{y}^j$. Its lowest nonzero total degree is the multiplicity. A smooth point of a variety has multiplicity one; a double point has multiplicity two. On a point blowup of a smooth algebraic surface, the total transform is the strict transform of an algebraic subvariety plus this multiplicity times the algebraic exceptional divisor.

For a nonzero homogeneous polynomial of degree $d>0$, multiplication gives $0\to {\mathcal{O}}_{{\mathbb{P}}^2}(-d)\to {\mathcal{O}}_{{\mathbb{P}}^2}\to i_{\ast }{\mathcal{O}}_X\to 0$. Intermediate cohomology of the twisting sheaves vanishes. The top cohomology of $\mathcal{O}(-d)$ has Laurent basis $t_0^{-b_0}{t}_1^{-b_1}{t}_2^{-b_2}$ with $b_i\ge 1$ and $\sum b_i=d$. There are $(d-1)(d-2)/2$ such monomials, with zero basis for $d=1,2$. The long exact sequence gives the displayed dimensions. The calculation allows singular, reducible and nonreduced hypersurfaces; it is an arithmetic-genus calculation, not an assertion about a normalization's genus.

Two distinct singular points would impose at least two zeros each on the restriction of the cubic to their joining line. A nonzero homogeneous polynomial of degree three cannot have total zero multiplicity four, so that restriction vanishes identically. The line would be a component, contradicting the irreducible polynomial hypothesis. This argument works in every characteristic.

In its classical nine-point cubic form, the Cayley-Bacharach theorem says that a cubic through eight of the nine distinct intersections of two plane cubics must contain the ninth. The eight-point cubic completion for two triples of lines is the elementary case needed for triangular-strip propagation in incidence geometry. The phenomenon explains why interpolation conditions at such a complete intersection are dependent.

Let two triples of projective lines meet in nine distinct points. Any plane cubic containing eight contains all nine. To prove this, write the triples as $F=L_1{L}_2{L}_3$ and $G=M_1{M}_2{M}_3$, with the missing point on $L_3$. The candidate cubic $H$ agrees with a scalar multiple of $G$ on $L_1$, so $H-\lambda G=L_{1}Q$. Its three known zeros on $L_2$ force $Q=L_{2}L$. Two remaining known zeros on $L_3$ force the linear factor $L$ to be a multiple of $L_3$. Hence $H=\lambda G+\mu F$ and vanishes at the missing point.

A two-cell-thick strip of triangles with degree-six vertices in a dual arrangement of a planar point set produces three indexed primal point families with collinearities $a_i,b_j,c_k$ whenever $i+j+k=0$. A plane cubic can be fitted to nine initial points because its homogeneous coefficient space has dimension ten. Overlapping nine-point configurations then force each next point onto the same cubic by eight-point cubic completion for two triples of lines. Safe neighbourhoods guarantee the distinctness needed in these local completion steps. A possible seed is $a_{-1},a_0,a_1,a_2,b_{-3},b_{-2},b_{-1},c_1,c_2$. Adjacent completion blocks force $c_3,b_{-4},c_4,a_{-2}$, followed by the alternating continuation along the two long transverse families.

A smooth complex degree-two projective plane curve is a projective linear image of the degree-two Veronese map and is diffeomorphic to a sphere. Its class in ${\mathbb{CP}}^2$ is twice the class of a projective line. Its self-intersection number is four, so its oriented normal bundle has Euler class evaluating to four.

The tangent lines of a smooth plane conic form a smooth plane conic in the dual projective space. If the original equation is $v^{\mathsf{T}}Qv=0$, its tangent has coefficient vector $Qv$, giving the displayed equation. The matrix $Q$ is symmetric and invertible, so this correspondence is a projective linear isomorphism of the conics.

If two smooth plane conics over $\mathbb{C}$ meet at four distinct points, their tangent incidence curve is a smooth projective genus one curve. Projection to $C_2$ is a double cover branched at those four intersections. The tangent equation becomes a quadratic whose discriminant cuts out $C_1\cap C_2$; its four simple zeros make the cover smooth and connected. The Riemann-Hurwitz formula then gives genus one.

For two smooth plane conics meeting transversely, the operation of crossing a chord of the second conic tangent to the first and choosing the other tangent through the new endpoint acts as a translation on an elliptic curve on their incidence curve. Each switch is an involution of a degree-two map from a genus one curve. One periodic orbit means that the translating point is torsion, so every orbit is periodic with the same least period. The construction extends through coincident choices at ramification points using the regular involutions.

The complement of a closed tubular neighborhood of a smooth plane conic has the displayed integral homology. The Excision theorem and Thom isomorphism theorem reduce the calculation to the relative long exact sequence: the ambient degree-four fundamental class restricts with coefficient one, while the degree-two map is intersection with the conic and has coefficient two. A collar neighborhood identifies the open and compact exterior homotopy equivalence types. The boundary instead has first homology $\mathbb{Z}/4$, from the Gysin sequence and the normal Euler class four.

A plane cubic is a projective plane curve defined by a nonzero homogeneous polynomial of degree three. Its arithmetic genus is one by the genus-degree formula, and its Hilbert polynomial is $3m$. A singular cubic can have normalization of smaller genus; the Hilbert polynomial measures the arithmetic genus.

A projective line $L=0$ is contained in the zero locus of the nonzero cubic $L^3$. The union of three lines is given by their product, and a conic together with a line is likewise a plane cubic. Reducibility and repeated factors are permitted in polynomial coverings used in incidence geometry; a covering by cubics need not consist of smooth irreducible curves.

This smooth projective plane curve has no real rational points: a sum of three real squares is zero only when all three coordinates vanish, which is forbidden in projective space. It is therefore not isomorphic over $\mathbb{R}$ to the projective line, which has rational points. Over $\mathbb{C}$ it becomes a nonsingular conic isomorphic to a projective line. Thus the ground field changes whether a geometric genus-zero curve admits a projective-line parametrization.

The arithmetic genus of a degree-$d$ projective plane curve isEquivalently, its structure sheaf has Euler characteristic $\chi ({\mathcal{O}}_X)=1-p_a$.

Blow up each distinct singular point of an integral projective curve in the projective plane once. The arithmetic genus drop under a point blowup and the genus-degree formula give arithmetic genus $(d-1)(d-2)/2-{\sum }_P{m}_P(m_P-1)/2$ for the strict transform. Its arithmetic genus is nonnegative, proving the bound in every characteristic. Infinitely near singularities need not have been resolved.

Three distinct points of multiplicity two use the entire arithmetic genus three of an integral projective curve of degree four. One point blowup at each gives arithmetic genus zero. The normalization of an algebraic curve then has genus zero and is the projective line over an algebraically closed field, by rationality of a smooth projective genus-zero curve. This allows double points with one branch as well as ordinary nodes.

Intersection multiplicity measures the local order of contact of two algebraic subvarieties. Transverse intersections have multiplicity one.

A smooth point of an algebraic plane curve is a flex if its tangent line has intersection multiplicity at least three there. This definition also applies to smooth points of a singular curve.

For a plane curve defined by a homogeneous polynomial $F$ of degree $d\ge 2$, assume the field characteristic is different from two and $d-1$ is nonzero in the field. A smooth point of a variety is a flex exactly when the determinant of the homogeneous Hessian matrix vanishes there, provided its tangent line is not a component. Put the point at $[0:0:1]$ with tangent $y=0$ and $F_y(p)\ne 0$. The homogeneous identities give $F_{xz}(p)=F_{zz}(p)=0$ and $F_{yz}(p)=(d-1)F_y(p)$, so the determinant is $-(d-1{)}^2{F}_y(p{)}^2{F}_{xx}(p)$. In the completed local ring, write the curve as a formal power series $y=h(x)$. Its linear coefficient is zero and its quadratic coefficient is $-F_{xx}(p)/(2F_y(p))$. Vanishing of that coefficient is exactly intersection multiplicity at least three with the tangent, proving the criterion in these positive characteristics as well as in characteristic zero.

For a cubic homogeneous polynomial $F$, its Hessian matrix has linear entries, so its determinant is a cubic homogeneous polynomial, possibly zero. On a nonsingular plane cubic in characteristic different from two, its zeros are exactly the flexes, by the Hessian criterion for a flex. If the determinant is nonzero, Bézout's theorem supplies intersections; if it is zero, the local criterion itself supplies flexes.