The Stacks project

Lemma 53.16.3. Let $k$ be an algebraically closed field. Let $X$ be a reduced algebraic $1$-dimensional $k$-scheme. Let $x \in X$. The following are equivalent

  1. $x$ defines a multicross singularity,

  2. the $\delta $-invariant of $X$ at $x$ is the number of branches of $X$ at $x$ minus $1$,

  3. there is a sequence of morphisms $U_ n \to U_{n - 1} \to \ldots \to U_0 = U \subset X$ where $U$ is an open neighbourhood of $x$, where $U_ n$ is nonsingular, and where each $U_ i \to U_{i - 1}$ is the glueing of two points as in Example 53.15.2.

Proof. The equivalence of (1) and (2) is Lemma 53.16.1.

Assume (3). We will argue by descending induction on $i$ that all singularities of $U_ i$ are multicross. This is true for $U_ n$ as $U_ n$ has no singular points. If $U_ i$ is gotten from $U_{i + 1}$ by glueing $a, b \in U_{i + 1}$ to a point $c \in U_ i$, then we see that

\[ \mathcal{O}_{U_ i, c}^\wedge \subset \mathcal{O}_{U_{i + 1}, a}^\wedge \times \mathcal{O}_{U_{i + 1}, b}^\wedge \]

is the set of elements having the same residue classes in $k$. Thus the number of branches at $c$ is the sum of the number of branches at $a$ and $b$, and the $\delta $-invariant at $c$ is the sum of the $\delta $-invariants at $a$ and $b$ plus $1$ (because the displayed inclusion has codimension $1$). This proves that (2) holds as desired.

Assume the equivalent conditions (1) and (2). We may choose an open $U \subset X$ such that $x$ is the only singular point of $U$. Then we apply Lemma 53.15.4 to the normalization morphism

\[ U^\nu = U_ n \to U_{n - 1} \to \ldots \to U_1 \to U_0 = U \]

All we have to do is show that in none of the steps we are squishing a tangent vector. Suppose $U_{i + 1} \to U_ i$ is the smallest $i$ such that this is the squishing of a tangent vector $\theta $ at $u' \in U_{i + 1}$ lying over $u \in U_ i$. Arguing as above, we see that $u_ i$ is a multicross singularity (because the maps $U_ i \to \ldots \to U_0$ are glueing of pairs of points). But now the number of branches at $u'$ and $u$ is the same and the $\delta $-invariant of $U_ i$ at $u$ is $1$ bigger than the $\delta $-invariant of $U_{i + 1}$ at $u'$. By Lemma 53.16.1 this implies that $u$ cannot be a multicross singularity which is a contradiction. $\square$

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