Degree Set of a Curve Analysis

• Degree set of a curve is the collection of degrees of closed points, defined via multiplicities and residue degrees from a smooth, proper, geometrically irreducible curve.
• This approach uses the combinatorial structure of a strict normal crossings divisor to extend the classical index computation to a complete description of all possible degrees.
• The algorithm leverages proper regular models over Henselian discrete valuation rings, linking geometric configurations (multiplicities and intersection matrices) with arithmetic invariants.

The degree set of a smooth, proper, geometrically irreducible algebraic curve $X$ defined over the field of fractions $K$ of a Henselian discrete valuation ring $R$ encodes the set of degrees of all closed points of $X$. Recent advances establish that this set can be computed entirely from the combinatorial data of the special fiber of a regular model of $X$ over $R$, provided the special fiber is a strict normal crossings divisor. This approach generalizes the classical computation of the index of $X$—the greatest common divisor (gcd) of degrees of closed points—by Gabber, Liu, and Lorenzini, to a complete description of all possible degrees realized on $X$ over $K$ (Creutz et al., 2023).

1. Basic Definitions

Let $R$ be a Henselian discrete valuation ring with field of fractions $K$ and residue field $k$. Let $X/K$ be a smooth, proper, geometrically irreducible curve.

• For any closed point $x \in X$, the degree is $\deg(x) := [\kappa(x):K]$.
• The degree set is $$S(X) := \{\deg(x): x\text{ a closed point of } X\} \subset \mathbb{N}_{>0}$$.
• The index of $X$ is $\mathrm{ind}(X) := \gcd S(X)$.

A proper regular model of $X$ over $R$ is a flat, proper $R$-scheme $\mathcal{X}$ whose generic fiber $\mathcal{X}_K$ is isomorphic to $X$ and which is regular as a scheme. Its special fiber $\mathcal{X}_k$ is a divisor on $\mathcal{X}$ over $k$.

A special fiber is a strict normal crossings divisor (SNCD) if, étale-locally on $\mathcal{X}$, it is defined by $t_1 \cdots t_r = 0$, where $\{t_1, \ldots, t_r\}$ is a part of a regular system of parameters. Equivalently, the irreducible components $C_i$ are smooth over $k$, meet transversely, and no three components meet in a common point unless forced by dimension.

2. Main Combinatorial Description

Suppose $\mathcal{X} \to \mathrm{Spec}(R)$ is a proper regular model with special fiber

$\mathcal{X}_k = \sum_{i=1}^r m_i C_i$

where each $C_i$ is a smooth $k$-curve and $m_i$ is its multiplicity. For each closed point $x \in \mathcal{X}_k(\bar{k})$, define the sub-semigroup $N(x) \subset \mathbb{N}$ generated by $\{m_i : x \in C_i\}$.

The degree set of $X$ over $K$ is given by

$$S(X) = \bigcup_{x \in \mathcal{X}_k \text{ closed}} \deg_k(x) \cdot N(x)$$

where $\deg_k(x) = [\kappa(x):k]$. Thus, the elements of $S(X)$ are all products of residue field degrees and integer combinations of $m_i$ for components containing $x$.

The index recovers as $\mathrm{ind}(X) = \gcd(m_1,\ldots,m_r)$. This is a specialization of the general result of Gabber–Liu–Lorenzini.

Alternatively, $S(X)$ can be described via the intersection matrix $(C_i \cdot C_j)_{i,j}$:

• Write $a = (a_1, \ldots, a_r) \in \mathbb{Z}_{\geq 0}^r$.
• There is a nonempty effective divisor supported on the $C_i$ with coefficients $a$ if and only if for all $j$,

$\sum_{i=1}^r a_i (C_i \cdot C_j) \geq 0.$

Therefore,

$$S(X) = \bigcup_{\substack{a \in \mathbb{Z}_{\geq 0}^r \ \sum_i a_i (C_i \cdot C_j) \geq 0\ \forall j}} \left\{ \sum_{i=1}^r a_i m_i \right\}.$$

3. Algorithmic Determination from the Special Fiber

Given $(C_i, m_i)$ and intersection numbers $C_i \cdot C_j$:

1. List components $C_1,\ldots,C_r$ and multiplicities $m_1,\ldots,m_r$.
2. Compute all intersection numbers $I_{ij} := C_i \cdot C_j$.
3. For each closed point $x$ (classified by intersection type), compute $$N(x) = \langle m_i : x \in C_i \rangle \subset \mathbb{N}$$ and record $\deg_k(x)$.
4. The degree set is $S(X) = \bigcup_x \deg_k(x) \cdot N(x)$.

In practice, this involves analyzing points according to which subset of components they lie on, forming semigroups from the corresponding multiplicities, and taking the union after multiplication by the corresponding residue field degrees.

4. Illustrative Examples and Comparative Behavior

Example 1 (Genus 2 over $p$-adic field):

Let $K$ be a $p$-adic field with uniformizer $\pi$ and $\alpha \in R^\times$ with $\alpha^2-4$ not a square. For $C: y^2 = x^6 + \alpha\pi x^3 + \pi^2$, the minimal regular model's special fiber is of type IV (Namikawa–Ueno): three components with multiplicities 1, 2, 3 meeting in a chain.

• $k$-points on the 2-component give degrees in $2\mathbb{N}$, on the 3-component in $3\mathbb{N}$.
• There are no degree 1 points on the multiplicity 1 component since $C(K) = \emptyset$.
• The degree set $S(C) = 2\mathbb{N} \cup 3\mathbb{N}$. Thus, although $\gcd S(C)=1$ (index 1), degrees coprime to $6$ never occur.

Example 2 (Curves over finitely generated fields):

For fields $F$ finite or finitely generated, if $D/F$ has index $\delta$,

$\bigcup_{d \gg 0} d \cdot \delta \subset S(D),$

i.e., all sufficiently large multiples of the index occur as degrees. The above $p$-adic example demonstrates failure of this property over Henselian fields.

5. Parameters Governing Possible Degree Sets

For smooth curves of fixed genus $g\ge 2$ over $K$ Henselian with $k$ finite or algebraically closed, the possible degree sets $S(C)$ are highly constrained: only finitely many occur as $C$ varies. In genus 2 over a $p$-adic field or with finite residue field, $S(C)$ ranges among $\mathbb{N}$, $2\mathbb{N}$, $\mathbb{N}_{>1}$, $2\mathbb{N} \cup 3\mathbb{N}$, or unions with finite initial pieces coming from regular genus 2 fiber reductions.

6. Theoretical and Algorithmic Significance

The ability to compute not only the index but the full degree set from the combinatorics of the special fiber in a regular SNCD model over a Henselian base links the geometry of the special fiber to arithmetic data on the generic fiber. This contrasts sharply with the behavior over global fields, underscoring the profound arithmetic implications of the Henselian condition. The algorithm formalized in (Creutz et al., 2023) provides explicit computational access and facilitates classification results for degree sets in various families of curves.

• Creutz–Viray, "Degrees of points on varieties over Henselian fields" (Creutz et al., 2023)
• Gabber–Liu–Lorenzini, "The index of an algebraic variety," Invent. Math. (2013)