For each value of \(d=2,3,4,5\) and \(r\) satisfying \(0 \leq r \leq \frac{1}{2}(d-1)(d-2)\), show that there exists an irreducible plane curve of degree \(d\) with \(r\) nodes and no other singularities.
Solution
Put \[ \delta_d=\frac{(d-1)(d-2)}2. \]
1. For every \(d\geq3\) there is an integral plane curve \(C_0\) of degree\(d\) having exactly \(\delta_d\) nodes and no other singularities.
Let \[ R_d\subseteq\PP^d \] be the degree-\(d\) rational normal curve. For \(d>3\), repeatedly apply Exercise IV.3.11(a) to project from a point outside the secant and tangent loci until \(R_d\) is embedded in \(\PP^3\). Each projection is a closed immersion on the curve and pulls the hyperplane bundle back to the preceding hyperplane bundle, so the resulting smooth rational space curve still has degree \(d\). For \(d=3\) we start directly with the twisted cubic.
The generic projection theorem for curves now gives a general projection of this space curve to an integral plane curve \[ C_0\subseteq\PP^2 \] whose only singularities are nodes and whose normalization is \(\PP^1\). Projection preserves the hyperplane degree, so \(\deg C_0=d\). If \(s\) is the number of nodes, the genus-drop formula gives \[ 0 = g(\PP^1) = \frac{(d-1)(d-2)}2-s. \] Hence \[ s=\delta_d. \]
2. Let \(C\subseteq\PP^2\) be any integral degree-\(d\) curve having exactly\(\delta\) nodes and no other singularities, and let\[ N=\{P_1,\ldots,P_\delta\} \]be its reduced node subscheme. Then\[ H^1\bigl(\PP^2,\mathcal I_N(d)\bigr)=0. \]
Let \[ \nu:\widetilde C\longrightarrow C \] be the normalization, and write \[ \nu^{-1}(P_i)=\{P_i',P_i''\}, \qquad \widetilde N=\sum_{i=1}^{\delta}(P_i'+P_i''). \] Put \[ L = \nu^*\mco_C(d)(-\widetilde N). \]
If \(G\) is a degree-\(d\) form vanishing at every node, then its restriction to \(C\), pulled back to \(\widetilde C\), vanishes at both branches over every node. Thus restriction gives an injective map \[ \frac{H^0(\PP^2,\mathcal I_N(d))}{kF} \hookrightarrow H^0(\widetilde C,L), \] where \(F\) is an equation of \(C\). The kernel is precisely \(kF\), since an additional degree-\(d\) form vanishing identically on the integral degree-\(d\) curve \(C\) is a scalar multiple of \(F\).
The normalization has genus \[ g = \frac{(d-1)(d-2)}2-\delta, \] while \[ \deg L=d^2-2\delta. \] Consequently \[ \deg L-(2g-2)=3d>0, \] so \(L\) is nonspecial. Riemann–Roch gives \[ \begin{aligned} h^0(\widetilde C,L) &=d^2-2\delta+1-g\\ &=\frac{d(d+3)}2-\delta. \end{aligned} \]
Set \[ \sigma=h^1(\PP^2,\mathcal I_N(d)). \] From \[ 0\longrightarrow\mathcal I_N(d) \longrightarrow\mco_{\PP^2}(d) \longrightarrow\mco_N \longrightarrow0 \] and \(H^1(\PP^2,\mco(d))=0\), we obtain \[ h^0(\PP^2,\mathcal I_N(d)) = \binom{d+2}{2}-\delta+\sigma. \] The injection above therefore yields \[ \frac{d(d+3)}2-\delta+\sigma \le \frac{d(d+3)}2-\delta. \] Hence \(\sigma=0\), as claimed.
3. The nodes of \(C\) can be smoothed independently inside the completelinear system of plane curves of degree \(d\).
The vanishing in step 2 makes the evaluation map \[ H^0(\PP^2,\mco(d)) \longrightarrow H^0(N,\mco_N) \cong k^\delta \] surjective.
We identify this map with the differentials of the local node-smoothing parameters. If \(F=0\) is the equation of \(C\) and \(G\) is a degree-\(d\) form, then the first-order deformation \[ F+\varepsilon G=0 \] has, at a node \(P_i\), local deformation class equal to \[ G(P_i) \in \frac{\mco_{\PP^2,P_i}} {(F,F_x,F_y)} \cong k. \] Indeed an ordinary node is, after an étale change of local coordinates, the germ \(uv=0\), whose one-dimensional smoothing space has parameter in \(uv=\tau_i\). Modulo the Jacobian ideal, a perturbing function contributes exactly its constant value at the node.
Thus, étale-locally at \([C]\) in the projective space of degree-\(d\) equations, there is a map \[ (\tau_1,\ldots,\tau_\delta) : \mathcal U\longrightarrow\AA^\delta \] whose \(i\)th coordinate is the smoothing parameter of \(P_i\), and whose differential is the displayed evaluation map. That differential is surjective, so the Jacobian criterion makes this map smooth at \([C]\). After shrinking \(\mathcal U\), every prescribed pattern \[ \tau_i=0 \quad\text{or}\quad \tau_i\ne0 \] therefore occurs.
For \(\tau_i=0\), the nearby singularity remains an ordinary node; for \(\tau_i\ne0\), it is smoothed. Nodality is an open condition along the zero-parameter locus. Since \(C\) has no singular points away from the \(P_i\), after shrinking once more the nearby curves acquire no new singularities elsewhere. Finally, geometric integrality is open in this flat projective family, and \(C\) itself is integral. Hence the deformation may be chosen integral while smoothing exactly any prescribed subset of the nodes and retaining all the others as nodes.
4. For every \(d\geq3\) and every integer\[ 0\le r\le\delta_d, \]there is an integral plane curve of degree \(d\) with exactly \(r\) nodes and noother singularities.
Start with the curve \(C_0\) from step 1, whose node set has cardinality \(\delta_d\). Choose any subset of \(r\) nodes to retain. By step 3 there is an integral degree-\(d\) deformation in which precisely those \(r\) local smoothing parameters remain zero and every other one is nonzero. The resulting curve has exactly the chosen \(r\) nodes and no other singularities.
5. The assertion also holds for \(d=2\).
Here \[ \delta_2=0. \] A nonsingular plane conic is irreducible and has no singularities, so it is the required curve for the only possible value \(r=0\).
6. Q.E.D.
For \(d=3,4,5\), step 4 supplies the required curve for every \[ 0\le r\le\frac{(d-1)(d-2)}2. \] Step 5 handles \(d=2\).