Projective Plane Curves

Abstract

Let \(X\subset \mathbb {P}^2\) be a divisor of degree d, with equation \(f(x_0,x_1,x_2)=0\), where f is a homogeneous polynomial of degree d, which we will call a projective plane curve or simply a curve. Recall from Sect. 1.6.5 that if we have the decomposition in distinct irreducible components $$ f=f_1^{h_1}\cdots f_n^{h_n} $$ then the curves \(X_i=Z_p(f_i)\), \(i=1,\ldots , n\), are called the irreducible components of X and one writes \(X=\sum _{i=1}^n h_iX_i\), where \(h_i\) is called the multiplicity of \(X_i\) in X, for \(i=1,\ldots , n\). Recall form Exercises 14.1.10 and 14.1.11 that if P is a point of X we defined the multiplicity \(m_P(X)\) of P for X. We defined also the tangent cone \(TC_{X,P}\) of X at P. This is the union of \(m=m_P(X)\) lines through P, each counted with a certain multiplicity, that are called the principal tangent lines to X at P.