A new preprint proposes an exact finite way to decide a question otherwise framed through an infinite sequence of algebraic operations: whether the Cox ring of a toric point blow-up is generated in Rees multiplicity one. In practical terms, the result identifies when the construction needs no generator from a higher Rees multiplicity.
The work is theoretical rather than experimental. It studies complete toric varieties, Cox rings, lattice ideals and associated Rees algebras over an algebraically closed field of characteristic zero. The authors’ main tool is a finite criterion built from projected lattice ideals, their analytic spreads, and whether a particular product of Cox variables is a zero divisor in the associated graded ring.
Turning an infinite question into finite checks
The central equivalence compares two conditions. For every nonempty set of projected coordinates with positive associated lattice data, the analytic spread must be no more than the number of selected coordinates minus one. This finite boundary test is equivalent to a dimension condition in the associated graded ring, which records the successive layers of the ideal.
That numerical test is not, by itself, the whole conclusion in the general setting. If the class of the product of the Cox variables is not a zero divisor in the associated graded ring, the saturated and ordinary extended Rees algebras coincide, and multiplicity-one generation follows. When the associated graded ring is unmixed, the finite numerical conditions imply this non-zero-divisor property and therefore give the generation result.
The distinction matters because the finite numbers alone do not rule out lower-dimensional associated primes on the coordinate boundary. The paper therefore treats the zero-divisor or unmixedness requirement as the extra algebraic safeguard needed to turn the finite calculation into a generation theorem.
Exact answers in lower-dimensional settings
For fake weighted projective spaces of dimension at least two, the paper obtains a clean equivalence: Rees multiplicity-one generation occurs exactly when the toric-point ideal is a complete intersection. That turns the generation question into a structural test on the ideal itself.
The surface result is similarly concrete. For projective toric surfaces, multiplicity-one generation is equivalent to a local test involving every positive three-coordinate projection. It is also equivalent to generation after every corresponding three-ray contraction, tying the algebraic property to a set of toric contractions.
A further corollary covers every minimal complete toric surface of Picard rank two: all such surfaces satisfy Rees multiplicity-one generation. In the supporting algebra, codimension-two radical lattice ideals have analytic spread two when they are complete intersections and three otherwise; over an infinite field, the relevant Rees algebra and associated graded ring are Cohen–Macaulay.
What the finite enumeration found
The authors then apply the criteria to a finite three-dimensional family of 225 canonical Fano tetrahedra. The enumeration reports 131 cases reduced by automorphisms, leaving 94 cases for the remaining analysis. Among those, 27 have complete-intersection ideals, while 67 do not have multiplicity-one generation. Across the full set, 158 blow-ups are reported as generated in multiplicity one.
A named obstruction in the surface case
The paper identifies an exact obstruction for toric partial resolutions of Gorenstein toric Fano surfaces. Multiplicity-one generation holds if and only if the surface does not dominate a particular exceptional rank-one surface, denoted X_exc.
For that exceptional surface, the authors give an explicit decomposition of the saturated powers of its toric-point ideal. Its Cox ring is generated by the base ring, t, the ideal in Rees degree one, and an additional element written as g t^-2. That element is necessary, making two the first additional Rees multiplicity.
Scope of the result
The authors present the analytic-spread and zero-divisor framework as a finite solution to the saturation problem for the toric settings covered by the paper. The classifications are conditional on the stated varieties, lattice ideals, and field assumptions, and the work does not establish an analogous criterion for all higher-dimensional projective toric varieties.
The manuscript is an arXiv preprint, version 1, dated 26 Aug 2026. The authors report partial support from Proyecto FONDECYT Regular No. 1230287 and state that the Magma code used for the computations is available at github.com/alaface/multiplicity-one.
Paper data and sources
Original title: Multiplicity-One Cox Rings of Toric Point Blow-Ups
Authors: Antonio Laface, Luca Ugaglia
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-08-26
DOI: Not available
Original paper · Full text