(Delta invariant)
Let \(p\) be a singular point of a curve \(C\) with normalisation \(\nu : \tilde C \to C\). Then \[ \delta_p \da \dim_k \qty{ \qty{\nu_* \OO_{\tilde C} / \OO_C}_p } , \] a finite number, and \(r_p\) denotes the number of analytic branches of \(C\) at \(p\).
(Node, cusp, and tacnode)
Up to analytic isomorphism at the origin:
| name | equation | \(r_p\) | \(\delta_p\) |
|---|---|---|---|
| node, \(A_1\) | \(y^2 = x^2\) | \(2\) | \(1\) |
| cusp, \(A_2\) | \(y^2 = x^3\) | \(1\) | \(1\) |
| tacnode, \(A_3\) | \(y^2 = x^4\) | \(2\) | \(2\) |
A node is a singular point of multiplicity \(2\) with two distinct tangent directions. More generally \(A_n : y^2 = x^{n+1}\) has \(\delta_p = \floor{(n+1)/2}\), and an ordinary point of multiplicity \(r\), meaning \(r\) smooth branches with distinct tangents, has \(\delta_p = \binom{r}{2}\).
(Genus drop)
\[ p_a(C) - g(\tilde C) = \sum_{p \in \Sing C} \delta_p . \] For a plane curve of degree \(d\) this reads \(g = \binom{d-1}{2} - \sum_p \delta_p\).
(The genus formula with multiplicities)
Let \(C \subseteq \PP^2\) be an integral curve of degree \(d\). Resolve its singularities by successively blowing up singular points of the strict transforms, and let \(m_i\) run over the multiplicities of all singular points met in this process, including the infinitely near points on the strict transforms. Then \[g(\tilde{C}) = \frac{1}{2}(d-1)(d-2) - \frac{1}{2} \sum_i m_i(m_i - 1).\] If every singular point of \(C\) is ordinary, no infinitely near singular points occur, and the sum runs over the singular points of \(C\).
The tacnode \(y^2 = x^4\) has multiplicity \(2\) at the origin and \(\delta = 2\). One blowup, in the chart \(y = x y_1\), gives the strict transform \(y_1^2 = x^2\), which is a node at the origin: an infinitely near singular point of multiplicity \(2\). The two points contribute \(\frac{1}{2}(2 \cdot 1 + 2 \cdot 1) = 2 = \delta\), while the origin alone would give \(1\).
(Invariants separating the normal forms)
Node and cusp both have \(\delta_p = 1\) and are separated by the branch count: \(\nu^{-1}(p)\) has two points for the node and one for the cusp. A nodal and a cuspidal plane cubic both have geometric genus \(0\), and their groups \(\Pic^0\) are \(\GG_m\) and \(\GG_a\) respectively (FE-ADOKK).
Node and tacnode both have two smooth branches. For a singular point with branches \(B_1,\ldots,B_r\), \(\delta_p=\sum_i\delta_p(B_i)+\sum_{i<j}(B_i\cdot B_j)_p\); for two smooth branches this is their intersection multiplicity, which is \(1\) for the transverse branches \(y=\pm x\) of the node and \(2\) for the tangent branches \(y=\pm x^2\) of the tacnode.
The Milnor number is \(\mu_p = 2\delta_p - r_p + 1\): the node and the cusp have \(\mu = 1, 2\) and the tacnode \(\mu = 3\), which is the \(n\) in \(A_n\).
(Singularities of general projections)
By T-CRVEMBP3, projection of a smooth curve in \(\PP^3\) from a general point is birational onto a plane curve whose only singularities are nodes. The centres of projection that produce a cusp lie on a tangent line of the curve, and those that produce a tacnode lie on a secant line whose two tangent lines are coplanar; each set has dimension less than \(3\), so a general centre avoids them.