Let \(Y \subseteq \AA^2\) be a curve defined by the equation \(f(x,y) = 0\), and let \(P = (a,b)\) be a point of \(\AA^2\). Make a linear change of coordinates so that \(P\) becomes the point \((0,0)\). Write \(f\) as a sum \(f = f_0 + f_1 + \cdots + f_d\), where \(f_i\) is homogeneous of degree \(i\) in \(x\) and \(y\). Define the multiplicity of \(P\) on \(Y\), denoted \(\mu_P(Y)\), to be the least \(r\) with \(f_r \neq 0\). Note that \(P \in Y\) if and only if \(\mu_P(Y) > 0\). The linear factors of \(f_r\) are called the tangent directions at \(P\).
Show that \(\mu_P(Y) = 1\) if and only if \(P\) is a nonsingular point of \(Y\).
Find the multiplicity of each of the singular points of the four quartics \(x^2 = x^4 + y^4\), \(xy = x^6 + y^6\), \(x^3 = y^2 + x^4 + y^4\), and \(x^2 y + x y^2 = x^4 + y^4\).
Solution
Translate coordinates so that the point under consideration is the origin, as in the statement. Then \(P\in Y\) means \(f_0=0\).
1. The linear homogeneous part of the translated equation is \[ f_1=f_x(P)x+f_y(P)y. \]
Write the original coordinates as \(X=a+x\) and \(Y=b+y\). For a monomial \(X^iY^j\), the terms of total degree one after substitution are \[ i a^{i-1}b^j x+j a^i b^{j-1}y. \] Summing over the monomials of \(f\) gives exactly the displayed expression. This computation is polynomial and is valid in every characteristic.
2. One has \[ \boxed{\mu_P(Y)=1\quad\Longleftrightarrow\quad P\text{ is nonsingular on }Y.} \]
Because \(P\in Y\), the multiplicity is one exactly when \(f_1\ne0\). By step 1 this is equivalent to \[ (f_x(P),f_y(P))\ne(0,0). \] For a plane hypersurface, the Jacobian criterion says precisely that \(P\) is nonsingular when at least one first partial derivative is nonzero [@Har10a, Chapter I, §5]. This proves part (1).
3. At the origin, the multiplicities of the four curves from Exercise I.5.1 are \[ \boxed{2,\quad2,\quad2,\quad3} \] in the order listed.
The defining polynomials, grouped by degree, begin as \[ \begin{aligned} f_1&=x^2-(x^4+y^4),\\ f_2&=xy-(x^6+y^6),\\ f_3&=-y^2+x^3-(x^4+y^4),\\ f_4&=(x^2y+xy^2)-(x^4+y^4). \end{aligned} \] Their first nonzero homogeneous pieces have degrees \(2,2,2,3\), respectively. By the definition in the statement, these degrees are exactly the four multiplicities. The corresponding tangent cones are \[ x^2,\qquad xy,\qquad -y^2,\qquad xy(x+y), \] agreeing with the tacnode, node, cusp, and ordinary triple point identified in P-AGH51PLANECURVESING.
4. In characteristics \(7\) and \(13\), the two additional singular points of the third curve also have multiplicity \(2\).
By P-AGH51PLANECURVESING, these points are \[ P_\pm=\left(\frac34,\ \pm\sqrt{-\frac12}\right). \] Their first-order terms vanish because they are singular. For \[ f_3=x^3-y^2-x^4-y^4, \] the second partial derivatives are \[ (f_3)_{xx}=6x-12x^2,\qquad (f_3)_{xy}=0,\qquad (f_3)_{yy}=-2-12y^2. \] At either \(P_\pm\) they become \[ (f_3)_{xx}=-\frac94,\qquad (f_3)_{yy}=4. \] These values are nonzero in characteristics \(7\) and \(13\). Hence the translated equation has a nonzero quadratic part and no nonzero term of lower degree, so its multiplicity is \(2\).