Let \(Y \subseteq \PP^2\) be a curve of degree greater than one. Regard the set of lines in \(\PP^2\) as another projective space \(\dualof{(\PP^2)}\), taking \((a_0, a_1, a_2)\) as homogeneous coordinates of the line \[ L: a_0 x_0 + a_1 x_1 + a_2 x_2 = 0. \]
For each nonsingular point \(P \in Y\), show that there is a unique line \(T_P(Y)\) whose intersection multiplicity with \(Y\) at \(P\) is \(> 1\). This is the tangent line to \(Y\) at \(P\).
Show that the map \(P \mapsto T_P(Y)\) defines a morphism from \(\Reg Y\), the set of nonsingular points of \(Y\), into \(\dualof{(\PP^2)}\). The closure of the image of this morphism is called the dual curve \(\dualof{Y} \subseteq \dualof{(\PP^2)}\) of \(Y\).
(The line case)
Exercise I.7.3 is stated for an arbitrary plane curve, but its reference to the intersection multiplicity of Exercise I.5.4 requires the two curves to be distinct. If \(Y\) is a line, its tangent line at every point is \(Y\) itself, so \((Y.Y)_P\) is not defined by I.5.4. The gradient formula in step <1>4 still defines the constant tangent map, whose image is the single point of \(\dualof{(\PP^2)}\) representing \(Y\). The degree-greater-than-one hypothesis above is therefore exactly what is needed for the exercise’s literal intersection-multiplicity formulation.
Solution
Let \(F(x_0,x_1,x_2)\) be an irreducible homogeneous equation of degree \(d\) for \(Y\). Write \(F_i=\partial F/\partial x_i\).
1. At a nonsingular point \(P\in Y\), there is a unique tangent direction in the affine tangent plane.
Choose an affine chart containing \(P\), translate \(P\) to the origin, and let \[ f(x,y)=f_1(x,y)+f_2(x,y)+\cdots \] be a local equation of the curve, decomposed into homogeneous terms. Since \(P\) is nonsingular, Exercise I.5.3 gives \[ f_1\ne0. \] Thus the linear equation \(f_1=0\) defines a unique line through the origin. This is the unique tangent direction at \(P\).
2. A line \(L\) through a nonsingular point \(P\) has intersection multiplicity \((L.Y)_P>1\) if and only if its local linear equation is proportional to \(f_1\).
Choose affine linear coordinates \((u,v)\) centered at \(P\) with \(L\) given by \(v=0\). Because \(L\) is not a component of the irreducible curve \(Y\) unless \(Y=L\), the local intersection multiplicity is the order of vanishing of the restricted equation \[ f(u,0)\in k[u]_{(u)}. \] Equivalently, \[ (L.Y)_P=\operatorname{length}k[u]_{(u)}/(f(u,0)). \] This length is greater than one exactly when \(f(u,0)\) has no nonzero linear term.
The linear term of \(f(u,0)\) is the restriction of \(f_1\) to \(L\). It vanishes identically exactly when \(f_1\) is a scalar multiple of the equation \(v\) of \(L\). Thus \((L.Y)_P>1\) exactly for the unique line \(f_1=0\) from step 1.
3. For \(P=[p_0:p_1:p_2]\in\Reg Y\), the unique line of step 2 is\[ \boxed{ T_P(Y):F_0(P)x_0+F_1(P)x_1+F_2(P)x_2=0}. \]
At least one \(F_i(P)\) is nonzero because \(P\) is nonsingular, so the displayed equation defines a line. Euler’s identity for the homogeneous form \(F\) gives \[ \sum_{i=0}^2 p_iF_i(P)=dF(P)=0, \] so the line passes through \(P\); this remains valid when \(d=0\) in \(k\) because \(F(P)=0\).
On any affine chart, the linear part of the translated local equation of \(Y\) at \(P\) is obtained by evaluating the first derivatives of \(F\) at \(P\). Hence the displayed line is precisely the line defined by that nonzero linear part. Step 2 therefore shows that it is the unique line whose intersection multiplicity with \(Y\) at \(P\) is greater than one.
4. The assignment \(P\mapsto T_P(Y)\) is a morphism\[ \gamma:\Reg Y\dualof{\longrightarrow(\PP^2)}. \]
In dual homogeneous coordinates, step 3 gives \[ \gamma(P)=[F_0(P):F_1(P):F_2(P)]. \] The three \(F_i\) are homogeneous polynomials of the same degree \(d-1\). They have no common zero on \(\Reg Y\), by the Jacobian criterion for nonsingularity. Therefore these three homogeneous forms define a morphism from \(\Reg Y\) to the dual projective plane. Its value at each point is exactly the tangent line from step 3.
5. The closure of \(\gamma(\Reg Y)\) is the dual algebraic set \(\dualof{Y}\).
The image of the irreducible open subset \(\Reg Y\) is irreducible, and its closure in the projective plane \(\dualof{(\PP^2)}\) is therefore an irreducible closed subset. By definition this closure is the dual curve \(\dualof{Y}\).
6. Q.E.D.
Steps <1>1–<1>3 prove existence and uniqueness of the tangent line by intersection multiplicity, and step 4 proves that the tangent-line assignment is a morphism. Step 5 identifies its closure with the dual curve.