Degree of a projective curve

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 embedding line bundle has at least four independent global sections. If it is nonspecial, the Riemann-Roch theorem gives $h^0=6-g$, so $g\le 2$. If it is special, the Clifford inequality for curves gives $h^0\le 3$, a contradiction. The smooth projective curve need not span its full ambient projective space.

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.

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

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 $\chi ({\mathcal{O}}_X)=1-p_a$.

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

The Hilbert function of a finitely generated graded algebra agrees in all sufficiently large degrees with a unique polynomial. For a projective variety of dimension $r$, its leading term is $\mathrm{deg}(V)m^r/r!$.

For a coherent sheaf $\mathcal{F}$ of support dimension $d$ on a projective scheme and a fixed Cartier divisor $D$, $h^0(X,\mathcal{F}(mD))=O(m^d)$. Choose a sufficiently high ample divisor $A$ whose section avoids the associated points of $\mathcal{F}$ and for which $D+A$ is ample. Multiplication by the $m$th power of that section injects $\mathcal{F}(mD)$ into $\mathcal{F}(m(D+A))$. The ample Hilbert polynomial, together with Serre vanishing, bounds the latter section space by $O(m^d)$.

For a general linear form $\ell$ not vanishing identically on a projective curve, the exact sequenceshows that the hyperplane-section length is $P_C(m)-P_C(m-1)=\mathrm{deg}C$.

Adjoining projective coordinates that vanish identically does not change the homogeneous coordinate ring or Hilbert polynomial. A linear embedding of projective space therefore preserves the degree of an embedded variety.

Projective degree belongs to an embedding rather than to the abstract variety. A line and a smooth conic in ${\mathbb{P}}^2$ are both abstractly ${\mathbb{P}}^1$ but have degrees one and two.