Rational Triple Points in Singularity Theory

• Rational Triple Point is a normal rational surface singularity of multiplicity 3, defined via explicit determinantal models and cubic hypersurface projections.
• Its resolution employs weighted dual graphs and Newton theory, enabling clear classification and toric resolution strategies.
• The canonical trace ideal organizes a structured Ulrich-ideal theory, offering actionable insights into nearly Gorenstein conditions and multiplicity constraints.

In singularity theory, a rational triple point is a normal rational surface singularity of multiplicity $3$. In the notation of the recent literature, it is an RTP-singularity. The rationality condition is expressed by the vanishing of the geometric genus, while the multiplicity condition places the singularity immediately after rational double points in the multiplicity stratification of rational surface singularities. RTP-singularities admit explicit determinantal models in $\mathbb C^4$, can be realized as normalizations of certain nonisolated cubic hypersurface singularities in $\mathbb C^3$, and in dimension two they support a particularly rigid Ulrich-ideal theory centered on the canonical trace ideal (Altintas et al., 2013, Maeda et al., 17 Jul 2025).

1. Definition and basic invariants

Let $(S,0)$ be a normal surface singularity and let

$\pi:(\widetilde S,E)\to (S,0)$

be a resolution. The singularity is rational if

$H^1(\widetilde S,\mathcal O_{\widetilde S})=0.$

Artin’s criterion, as recalled in the RTP literature, states that $(S,0)$ is rational if and only if $p_a(Y)\le 0$ for every positive divisor $Y=\sum a_iE_i$ supported on the exceptional divisor. For rational singularities, the exceptional divisor in a resolution is a normal crossings divisor, each irreducible exceptional component $E_i$ is a smooth rational curve, and the dual graph is a tree. A rational triple point is precisely a rational surface singularity with multiplicity $\mathbb C^4$0 (Altintas et al., 2013).

For rational singularities, Artin’s divisor $\mathbb C^4$1 satisfies

$\mathbb C^4$2

where $\mathbb C^4$3 is the multiplicity, and the embedding dimension is $\mathbb C^4$4. Specializing to multiplicity $\mathbb C^4$5, an RTP has $\mathbb C^4$6 and embedding dimension $\mathbb C^4$7. In the two-dimensional commutative-algebraic setting used in the Ulrich-ideal classification, one works with a two-dimensional excellent normal local domain $\mathbb C^4$8 over an algebraically closed field of characteristic $\mathbb C^4$9, with

$\mathbb C^3$0

Such a singularity has minimal multiplicity and Cohen–Macaulay type

$\mathbb C^3$1

Thus a two-dimensional rational triple point is automatically a normal Cohen–Macaulay non-Gorenstein local domain of type $\mathbb C^3$2 (Maeda et al., 17 Jul 2025).

2. Classical classification and determinantal models

Artin classified the minimal resolution graphs of rational surface singularities of multiplicity $\mathbb C^3$3. In the notation used for the nonisolated-form construction, the resulting RTP families are

$\mathbb C^3$4

In the later Ulrich-ideal treatment, the same class is organized via Artin–Tyurina weighted dual graphs

$\mathbb C^3$5

Both descriptions encode the same basic fact: RTP-singularities are classified through explicit weighted trees attached to their minimal resolutions (Altintas et al., 2013, Maeda et al., 17 Jul 2025).

A central structural feature is that RTP-singularities are neither hypersurface singularities nor complete intersections. Tjurina’s construction realizes them as surfaces in $\mathbb C^3$6 defined by the maximal minors of $\mathbb C^3$7 matrices. The literature also uses Miranda’s triple-cover form, in which the defining ideal is again determinantal. Resolution-theoretically, the Tjurina modification replaces the singularity by a birational model with central fibre $\mathbb C^3$8, corresponding to a $\mathbb C^3$9-curve, and the residual singularities along $(S,0)$0 are rational double points. This explains the standard picture of an RTP resolution graph as a central $(S,0)$1-vertex with attached rational-double-point configurations (Altintas et al., 2013).

3. Nonisolated hypersurface forms and Newton theory

A distinctive contribution of the RTP literature is the construction of explicit nonisolated hypersurface singularities in $(S,0)$2 whose normalizations are the normal RTP-singularities. The starting point is the observation that any normal surface singularity is the normalization of a nonisolated hypersurface singularity. For RTPs, suitable projections of the determinantal equations in $(S,0)$3 produce cubic hypersurfaces in $(S,0)$4; the normalization map is the induced finite birational morphism from the normal RTP model to the projected hypersurface (Altintas et al., 2013).

The projected hypersurfaces again fall into the Artin families. Representative examples are

$(S,0)$5

$(S,0)$6

$(S,0)$7

together with explicit cubic equations for the $(S,0)$8-, $(S,0)$9-, $\pi:(\widetilde S,E)\to (S,0)$0-, $\pi:(\widetilde S,E)\to (S,0)$1-, $\pi:(\widetilde S,E)\to (S,0)$2-, and $\pi:(\widetilde S,E)\to (S,0)$3-families. These hypersurfaces are nonisolated, but their normalizations recover the normal RTP-singularities exactly. The construction is not purely Newton-polygon bookkeeping: the cubic

$\pi:(\widetilde S,E)\to (S,0)$4

has the same Newton polygon as the $\pi:(\widetilde S,E)\to (S,0)$5 model, yet its normalization is smooth and the equation is degenerate. The hypersurface models used for RTPs are therefore genuinely special (Altintas et al., 2013).

The Newton polygon of a nonisolated RTP form controls its toric resolution. Using Oka’s resolution process, one passes from the Newton polyhedron to a dual fan, then to regular subdivisions of two-dimensional cones, and finally to a weighted graph. After blowing down the resulting $\pi:(\widetilde S,E)\to (S,0)$6-curves, the graph coincides with Artin’s minimal resolution graph. The same work proves that the nonisolated forms and their normalizations are both Newton non-degenerate. This places RTP-singularities at a useful intersection of rational-surface-singularity theory and toric/Newton resolution methods, and the paper notes that these examples are apparently the first nonisolated hypersurface singularities in the literature for which Oka’s resolution process works (Altintas et al., 2013).

4. Canonical trace ideals and the Ulrich condition

For a Cohen–Macaulay local ring $\pi:(\widetilde S,E)\to (S,0)$7, an $\pi:(\widetilde S,E)\to (S,0)$8-primary ideal $\pi:(\widetilde S,E)\to (S,0)$9 is Ulrich if there exists a minimal reduction $H^1(\widetilde S,\mathcal O_{\widetilde S})=0.$0 such that

$H^1(\widetilde S,\mathcal O_{\widetilde S})=0.$1

and $H^1(\widetilde S,\mathcal O_{\widetilde S})=0.$2 is a free $H^1(\widetilde S,\mathcal O_{\widetilde S})=0.$3-module. The numerical criterion recalled in the RTP literature is

$H^1(\widetilde S,\mathcal O_{\widetilde S})=0.$4

which in dimension $H^1(\widetilde S,\mathcal O_{\widetilde S})=0.$5 becomes

$H^1(\widetilde S,\mathcal O_{\widetilde S})=0.$6

If $H^1(\widetilde S,\mathcal O_{\widetilde S})=0.$7 is generically Gorenstein with canonical module $H^1(\widetilde S,\mathcal O_{\widetilde S})=0.$8, the trace ideal

$H^1(\widetilde S,\mathcal O_{\widetilde S})=0.$9

is the canonical trace ideal, and the residue is

$(S,0)$0

A ring is nearly Gorenstein if

$(S,0)$1

For non-Gorenstein rational surface singularities, every Ulrich ideal contains the canonical trace ideal (Maeda et al., 17 Jul 2025).

For RTPs, the canonical trace ideal becomes the organizing object of the theory. If $(S,0)$2 is a non-Gorenstein Cohen–Macaulay local domain with $(S,0)$3, then $(S,0)$4, and there is a Hilbert–Burch resolution

$(S,0)$5

A key lemma identifies

$(S,0)$6

The main RTP theorem then states that for every two-dimensional rational triple point $(S,0)$7, the canonical trace ideal $(S,0)$8 is itself an Ulrich ideal. More precisely, there exist a minimal system of generators $(S,0)$9 of $p_a(Y)\le 0$0 and an integer $p_a(Y)\le 0$1 such that

$p_a(Y)\le 0$2

$p_a(Y)\le 0$3

$p_a(Y)\le 0$4

and $p_a(Y)\le 0$5 is nearly Gorenstein if and only if

$p_a(Y)\le 0$6

Thus, on an RTP, the canonical trace ideal is the minimal Ulrich ideal, and every Ulrich ideal is obtained by enlarging it (Maeda et al., 17 Jul 2025).

5. Explicit classification of Ulrich ideals

The family-wise classification over the Artin–Tyurina graphs shows that, in explicit coordinates, the canonical trace ideal always takes the form

$p_a(Y)\le 0$7

and the full set of Ulrich ideals is the chain

$p_a(Y)\le 0$8

This pattern is verified case by case for

$p_a(Y)\le 0$9

In each family the exponent $Y=\sum a_iE_i$0 is computed explicitly from the graph parameters, and the equality

$Y=\sum a_iE_i$1

is made completely explicit. The proofs use direct reductions $Y=\sum a_iE_i$2 and check

$Y=\sum a_iE_i$3

for $Y=\sum a_iE_i$4, which is exactly the Ulrich criterion in dimension $Y=\sum a_iE_i$5 (Maeda et al., 17 Jul 2025).

A representative example is the RTP $Y=\sum a_iE_i$6, for which

$Y=\sum a_iE_i$7

and these are exactly the two Ulrich ideals. More generally, the nearly Gorenstein RTPs are precisely those for which

$Y=\sum a_iE_i$8

The same paper establishes a sharp contrast with higher-multiplicity quotient singularities: if $Y=\sum a_iE_i$9 is a two-dimensional quotient singularity with multiplicity $E_i$0, then

$E_i$1

so the maximal ideal is the unique Ulrich ideal. It also proves

$E_i$2

for every two-dimensional quotient singularity. RTPs are therefore the multiplicity-$E_i$3 locus where nonmaximal Ulrich ideals still occur in a controlled way (Maeda et al., 17 Jul 2025).

6. Terminological scope and unrelated uses of “triple point”

The singularity-theoretic meaning of rational triple point should be separated from several unrelated uses of “triple point.” In low-dimensional topology, “triple point” may refer to triple points of filling Dehn surfaces and the triple point spectrum of a closed orientable $E_i$4-manifold, or to triple point numbers of surface-link diagrams (Rojo et al., 2014, Nakamura, 2011, Kharusi et al., 2017). In analysis, it can denote common boundary points of three domains in harmonic-measure theory (Tolsa et al., 2016). In operator algebras it refers to initial $E_i$5-valent branching in principal graphs of subfactors (Snyder, 2012). In condensed matter it denotes triply degenerate band crossings such as triple-point fermions in ferroelectric GeTe (Krempaský et al., 2020). In thermodynamics and adjacent analogical literature it denotes coexistence points in phase diagrams; the noble-gas work on critical and triple-point ratios explicitly does not propose a “rational triple point” as an exact simple fraction, and the Roegenian economics paper discusses an “economic triple point” only by analogy (Benilov et al., 2018, Udriste et al., 2018).

Within algebraic geometry and commutative algebra, however, the phrase has a precise and stable content: a rational surface singularity of multiplicity $E_i$6, equipped with explicit resolution graphs, determinantal equations, nonisolated hypersurface models, Newton non-degenerate realizations, and a canonical-trace-based Ulrich-ideal classification (Altintas et al., 2013, Maeda et al., 17 Jul 2025).