Mercat: Geometry, Discrete Analysis & Computation

• Mercat is a polysemous term defining several research strands including conjectures in higher-rank Clifford indices, discrete Riemann surfaces, arithmetic bounded partial quotients, and computational tools.
• It plays a crucial role in higher-rank Brill–Noether theory by framing conjectures like Mercat’s on rank-two Clifford indices and identifying failure loci for generic algebraic curves.
• In discrete complex analysis and machine learning, Mercat’s frameworks enable innovative methods such as quad-graph formulations and angle-preserving embeddings, inspiring tools like Exo-MerCat.

Searching arXiv for recent and foundational uses of “Mercat” to ground the article in the literature. Mercat is a surname and eponym that appears in several distinct research literatures. In current arXiv usage, it denotes, most notably, Mercat’s conjecture on higher-rank Clifford indices in algebraic geometry, the discrete complex-analytic framework initiated by C. Mercat on quad-graphs and discrete Riemann surfaces, and several unrelated contemporary computational systems, including the exoplanet catalog software Exo-MerCat and the angle-preserving embedding method Mercat (Bakker et al., 2015, Skopenkov, 2011, Alei et al., 2020, Fischer et al., 2024).

1. Mercat’s conjecture and higher-rank Clifford indices

In algebraic geometry, “Mercat” most commonly refers to a conjectural relation between the classical Clifford index of a smooth curve and its rank-$2$ analogues. For a smooth, irreducible algebraic curve $C$ of genus $g \geq 4$, the classical Clifford index is

$$\operatorname{Cliff}(C)=\min\{\deg L-2(h^0(L)-1)\mid L \text{ a line bundle},\ h^0(L)\ge 2,\ h^1(L)\ge 2\},$$

and one writes $\gamma_1=\operatorname{Cliff}(C)$ (Lange et al., 2010). For a semistable vector bundle $E$ of rank $n$,

$\gamma(E)=\frac{\deg E-2(h^0(E)-n)}{n},$

with associated higher-rank Clifford indices

$$\gamma_n:=\min\{\gamma(E)\mid E\text{ semistable rank }n,\ h^0(E)\ge n+1,\ \mu(E)\le g-1\},$$

and

$$\gamma_n':=\min\{\gamma(E)\mid E\text{ semistable rank }n,\ h^0(E)\ge 2n,\ \mu(E)\le g-1\}$$

(Lange et al., 2010).

Mercat’s conjecture for rank $C$0 predicts that the Clifford index for semistable rank-$C$1 bundles coincides with the classical Clifford index, namely

$C$2

(Lange et al., 2010). A closely related formulation defines, for rank $C$3,

$C$4

with

$C$5

and states that

$C$6

for a general curve $C$7 (Bakker et al., 2015).

The conjecture is structurally significant because it asks whether line bundles already detect the relevant Clifford-theoretic complexity of a curve, or whether semistable rank-$C$8 bundles can produce strictly smaller invariants. In the notation of Lange and Newstead, the rank-$C$9 invariants satisfy

$g \geq 4$0

where $g \geq 4$1 is the minimal degree of a line bundle with at least $g \geq 4$2 sections (Lange et al., 2010). This places Mercat’s conjecture squarely inside higher-rank Brill–Noether theory.

2. Validity, counterexamples, and failure loci

The status of Mercat’s conjecture is mixed: it is true for generic curves, but false in general. Lange and Newstead studied rank-$g \geq 4$3 vector bundles computing $g \geq 4$4 and $g \geq 4$5, with complete classifications in several curve classes, including smooth plane curves, exceptional curves, and $g \geq 4$6-gonal curves (Lange et al., 2010). Their main counterexample theorem states that if $g \geq 4$7 has Clifford dimension $g \geq 4$8 and genus $g \geq 4$9, then there exists a stable bundle $$\operatorname{Cliff}(C)=\min\{\deg L-2(h^0(L)-1)\mid L \text{ a line bundle},\ h^0(L)\ge 2,\ h^1(L)\ge 2\},$$0 of rank $$\operatorname{Cliff}(C)=\min\{\deg L-2(h^0(L)-1)\mid L \text{ a line bundle},\ h^0(L)\ge 2,\ h^1(L)\ge 2\},$$1, degree $$\operatorname{Cliff}(C)=\min\{\deg L-2(h^0(L)-1)\mid L \text{ a line bundle},\ h^0(L)\ge 2,\ h^1(L)\ge 2\},$$2, with $$\operatorname{Cliff}(C)=\min\{\deg L-2(h^0(L)-1)\mid L \text{ a line bundle},\ h^0(L)\ge 2,\ h^1(L)\ge 2\},$$3, and in particular

$$\operatorname{Cliff}(C)=\min\{\deg L-2(h^0(L)-1)\mid L \text{ a line bundle},\ h^0(L)\ge 2,\ h^1(L)\ge 2\},$$4

yielding infinitely many counterexamples to Mercat’s conjecture (Lange et al., 2010).

At the same time, the conjecture holds generically. The paper "The Mercat Conjecture for stable rank 2 vector bundles on generic curves" proves that for a general curve $$\operatorname{Cliff}(C)=\min\{\deg L-2(h^0(L)-1)\mid L \text{ a line bundle},\ h^0(L)\ge 2,\ h^1(L)\ge 2\},$$5 of genus $$\operatorname{Cliff}(C)=\min\{\deg L-2(h^0(L)-1)\mid L \text{ a line bundle},\ h^0(L)\ge 2,\ h^1(L)\ge 2\},$$6,

$$\operatorname{Cliff}(C)=\min\{\deg L-2(h^0(L)-1)\mid L \text{ a line bundle},\ h^0(L)\ge 2,\ h^1(L)\ge 2\},$$7

(Bakker et al., 2015). The proof uses moduli of sheaves on generic K3 surfaces, Lazarsfeld–Mukai bundles, and stability arguments on a polarized K3 surface $$\operatorname{Cliff}(C)=\min\{\deg L-2(h^0(L)-1)\mid L \text{ a line bundle},\ h^0(L)\ge 2,\ h^1(L)\ge 2\},$$8 with $$\operatorname{Cliff}(C)=\min\{\deg L-2(h^0(L)-1)\mid L \text{ a line bundle},\ h^0(L)\ge 2,\ h^1(L)\ge 2\},$$9 (Bakker et al., 2015).

The same work identifies a geometric failure locus in odd genus. For $\gamma_1=\operatorname{Cliff}(C)$0, the curves where the conjecture fails form an effective divisor

$\gamma_1=\operatorname{Cliff}(C)$1

and its slope is computed to be

$\gamma_1=\operatorname{Cliff}(C)$2

(Bakker et al., 2015). This places Mercat’s conjecture inside the birational geometry of $\gamma_1=\operatorname{Cliff}(C)$3: generic validity coexists with explicit divisorial failure.

3. Mercat in higher-rank Brill–Noether theory

Mercat’s influence extends beyond the conjecture itself. In the theory of twisted Brill–Noether loci, Mercat’s construction methods and stability criteria are used as foundational tools. For a smooth projective curve $\gamma_1=\operatorname{Cliff}(C)$4, a vector bundle $\gamma_1=\operatorname{Cliff}(C)$5, and integers $\gamma_1=\operatorname{Cliff}(C)$6, the twisted Brill–Noether locus is

$\gamma_1=\operatorname{Cliff}(C)$7

with twisted Brill–Noether number

$\gamma_1=\operatorname{Cliff}(C)$8

(Hitching et al., 2018).

Hitching–Hoff–Newstead describe their Theorem 1.1 as a direct generalization of Mercat’s theorem from the untwisted to the twisted setting. In their formulation, if $\gamma_1=\operatorname{Cliff}(C)$9, then for all $E$0, for all $E$1 (resp. $E$2), and for $E$3, the twisted locus $E$4 is nonempty (Hitching et al., 2018). Their smoothness analysis uses the twisted Petri trace map

$E$5

and smoothness at $E$6 is characterized by injectivity of this map (Hitching et al., 2018).

The same paper also shows that Mercat-type pathologies persist in the twisted setting. For general $E$7 and general $E$8, one can obtain twisted Brill–Noether loci that are nonempty even when the expected dimension is negative; for instance, when $E$9, $n$0, $n$1 is general of rank $n$2 and degree $n$3, and $n$4, the locus $n$5 has

$n$6

but is nonempty and has a component of dimension at least $n$7 (Hitching et al., 2018). In this literature, “Mercat” therefore designates a body of methods as well as a conjecture.

4. Mercat in discrete complex analysis and discrete Riemann surfaces

A second major use of the name concerns discrete complex analysis. Mercat was one of the researchers who introduced discrete analyticity on quad-graphs. In a finite planar graph whose bounded faces are quadrilaterals, a function $n$8 is called discrete analytic if, for each face $n$9,

$\gamma(E)=\frac{\deg E-2(h^0(E)-n)}{n},$0

(Skopenkov, 2011). This quad-graph Cauchy–Riemann condition became a central starting point for later work on discrete harmonic functions, discrete holomorphic functions, and discrete Riemann surfaces.

Skopenkov proved that the Dirichlet boundary value problem for the real part of a discrete analytic function has a unique solution on any finite quadrilateral lattice, and, when each face has orthogonal diagonals, that the solution uniformly converges to a harmonic function in the scaling limit (Skopenkov, 2011). Bobenko and Skopenkov then developed a linear discretization of complex analysis based on triangulations with cotangent weights, proving convergence of discrete period matrices and discrete Abelian integrals and establishing a discrete counterpart of the Riemann–Roch theorem (Bobenko et al., 2012).

Bobenko and Günther further generalized Mercat’s framework in two directions. First, for planar quad-graphs, they extended the linear theory from rhombic cases to arbitrary planar quad-graphs, giving discrete counterparts of holomorphic functions, derivatives, the Laplacian, exterior calculus, Green’s identities, and Cauchy’s integral formulae; they state that for the first time Green’s first identity and Cauchy’s integral formula for the derivative of a holomorphic function are discretized in this setting (Bobenko et al., 2015). Second, in the theory of discrete Riemann surfaces based on quadrilateral cellular decompositions, they moved from Mercat’s mainly real-weight setting to complex weights with $\gamma(E)=\frac{\deg E-2(h^0(E)-n)}{n},$1, where

$\gamma(E)=\frac{\deg E-2(h^0(E)-n)}{n},$2

and introduced discrete coverings, branched coverings, a discrete Riemann–Hurwitz formula, double poles of discrete one-forms, double values of discrete meromorphic functions in the discrete Riemann–Roch theorem, and a discrete Abel–Jacobi map (Bobenko et al., 2015). In this domain, “Mercat” identifies an originating discrete-analytic formalism rather than a single theorem.

5. Mercat in continued fractions and arithmetic

A third mathematical usage concerns continued fractions with bounded partial quotients. One strand of the literature attributes to Mercat a result in the number-field setting showing that Zaremba’s conjecture implies McMullen’s conjecture. The paper "Continued fractions in function fields: polynomial analogues of McMullen's and Zaremba's conjectures" translates this to function fields and proves a polynomial analogue of Mercat’s theorem (Malagoli, 2017). In the terminology of that paper, the polynomial analogue of Zaremba’s conjecture, $\gamma(E)=\frac{\deg E-2(h^0(E)-n)}{n},$3, asks for a constant $\gamma(E)=\frac{\deg E-2(h^0(E)-n)}{n},$4 such that for every nonconstant $\gamma(E)=\frac{\deg E-2(h^0(E)-n)}{n},$5 there exists $\gamma(E)=\frac{\deg E-2(h^0(E)-n)}{n},$6, coprime to $\gamma(E)=\frac{\deg E-2(h^0(E)-n)}{n},$7, with all partial quotients in the continued fraction of $\gamma(E)=\frac{\deg E-2(h^0(E)-n)}{n},$8 having degree $\gamma(E)=\frac{\deg E-2(h^0(E)-n)}{n},$9; the polynomial analogue of McMullen’s conjecture, $$\gamma_n:=\min\{\gamma(E)\mid E\text{ semistable rank }n,\ h^0(E)\ge n+1,\ \mu(E)\le g-1\},$$0, asks for a constant $$\gamma_n:=\min\{\gamma(E)\mid E\text{ semistable rank }n,\ h^0(E)\ge n+1,\ \mu(E)\le g-1\},$$1 such that for every suitable $$\gamma_n:=\min\{\gamma(E)\mid E\text{ semistable rank }n,\ h^0(E)\ge n+1,\ \mu(E)\le g-1\},$$2, there are infinitely many pairwise non-equivalent elements of $$\gamma_n:=\min\{\gamma(E)\mid E\text{ semistable rank }n,\ h^0(E)\ge n+1,\ \mu(E)\le g-1\},$$3 whose partial quotients have degree $$\gamma_n:=\min\{\gamma(E)\mid E\text{ semistable rank }n,\ h^0(E)\ge n+1,\ \mu(E)\le g-1\},$$4 (Malagoli, 2017).

The central implication is explicit: if Zaremba’s conjecture holds over $$\gamma_n:=\min\{\gamma(E)\mid E\text{ semistable rank }n,\ h^0(E)\ge n+1,\ \mu(E)\le g-1\},$$5 for constant $$\gamma_n:=\min\{\gamma(E)\mid E\text{ semistable rank }n,\ h^0(E)\ge n+1,\ \mu(E)\le g-1\},$$6 and $$\gamma_n:=\min\{\gamma(E)\mid E\text{ semistable rank }n,\ h^0(E)\ge n+1,\ \mu(E)\le g-1\},$$7 is Pellian, then there exist infinitely many quadratic irrationalities in $$\gamma_n:=\min\{\gamma(E)\mid E\text{ semistable rank }n,\ h^0(E)\ge n+1,\ \mu(E)\le g-1\},$$8 with partial quotients of degree at most $$\gamma_n:=\min\{\gamma(E)\mid E\text{ semistable rank }n,\ h^0(E)\ge n+1,\ \mu(E)\le g-1\},$$9 (Malagoli, 2017). The same paper connects this arithmetic to generalized Jacobians of hyperelliptic curves.

In a related direction, "Finiteness and periodicity of continued fractions over quadratic number fields" studies continued fractions with partial quotients in the ring of integers of a quadratic number field and uses a conjecture of Mercat as a boundary condition for finiteness phenomena (Masáková et al., 2019). There, Mercat’s conjecture is presented as the statement that every real quadratic field contains a periodic continued fraction with partial quotients equal to $$\gamma_n':=\min\{\gamma(E)\mid E\text{ semistable rank }n,\ h^0(E)\ge 2n,\ \mu(E)\le g-1\}$$0 or $$\gamma_n':=\min\{\gamma(E)\mid E\text{ semistable rank }n,\ h^0(E)\ge 2n,\ \mu(E)\le g-1\}$$1 (Masáková et al., 2019). The paper proves that for any quadratic Perron number $$\gamma_n':=\min\{\gamma(E)\mid E\text{ semistable rank }n,\ h^0(E)\ge 2n,\ \mu(E)\le g-1\}$$2, every element of $$\gamma_n':=\min\{\gamma(E)\mid E\text{ semistable rank }n,\ h^0(E)\ge 2n,\ \mu(E)\le g-1\}$$3 has a finite or eventually periodic $$\gamma_n':=\min\{\gamma(E)\mid E\text{ semistable rank }n,\ h^0(E)\ge 2n,\ \mu(E)\le g-1\}$$4-continued fraction, and that for four quadratic Perron numbers the $$\gamma_n':=\min\{\gamma(E)\mid E\text{ semistable rank }n,\ h^0(E)\ge 2n,\ \mu(E)\le g-1\}$$5-continued fraction represents finitely all elements of the quadratic field $$\gamma_n':=\min\{\gamma(E)\mid E\text{ semistable rank }n,\ h^0(E)\ge 2n,\ \mu(E)\le g-1\}$$6; based on the validity of Mercat’s conjecture, these are all quadratic Perron numbers with this feature (Masáková et al., 2019).

6. Unrelated contemporary systems bearing the name

The same string also appears in recent computational work with no direct relation to the algebraic-geometric or discrete-analytic literature. "Exo-MerCat: a merged exoplanet catalog with Virtual Observatory connection" defines Exo-MerCat as a Python code that collects and selects the most precise measurement for planetary and orbital parameters from four exoplanet databases—NASA Exoplanet Archive, Exoplanet Orbit Database, Exoplanet Encyclopaedia, and Open Exoplanet Catalogue—while handling aliases and notation differences, retrieving host-star information via Virtual Observatory ConeSearch, and providing a GUI for filtering and plotting (Alei et al., 2020). The catalog is available as a VO resource and is periodically updated (Alei et al., 2020).

The open-source release "Exo-MerCat v2.0.0" expands the system by adding the TESS Input Catalog and the K2 Input Catalog as input sources, optimizing main identifier queries, refining the merging logic, adding informative flags and log files, and refactoring the code into modules (Alei et al., 12 Feb 2025). The paper reports that the sample size almost doubled, from 9,900 planets in Exo-MerCat v1.1.0 to 17,445 planets in v2.0.0, with corresponding increases in confirmed and candidate entries (Alei et al., 12 Feb 2025).

A different recent use appears in machine learning. The paper "Sailing in high-dimensional spaces: Low-dimensional embeddings through angle preservation" introduces Mercat as a low-dimensional embedding method that reconstructs angles between data points rather than pairwise distances (Fischer et al., 2024). The method embeds data onto the $$\gamma_n':=\min\{\gamma(E)\mid E\text{ semistable rank }n,\ h^0(E)\ge 2n,\ \mu(E)\le g-1\}$$7-sphere $$\gamma_n':=\min\{\gamma(E)\mid E\text{ semistable rank }n,\ h^0(E)\ge 2n,\ \mu(E)\le g-1\}$$8 and minimizes a root mean square angle distortion over triplets,

$$\gamma_n':=\min\{\gamma(E)\mid E\text{ semistable rank }n,\ h^0(E)\ge 2n,\ \mu(E)\le g-1\}$$9

with optimization by gradient descent and computational accelerations via PCA denoising and triplet subsampling (Fischer et al., 2024). The name is explicitly stated to refer to the Mercator projection, with the intended analogy of angle preservation (Fischer et al., 2024).

Across these literatures, “Mercat” is therefore not a single concept but a polysemous research label. In algebraic geometry it chiefly designates a conjecture and its surrounding theory; in discrete complex analysis it denotes a foundational formalism on quad-graphs and discrete Riemann surfaces; in arithmetic it marks conjectures and implications about bounded partial quotients; and in recent computational work it names otherwise unrelated software and algorithms (Bakker et al., 2015, Bobenko et al., 2015, Malagoli, 2017, Fischer et al., 2024).