arXiv subjects

Maria Angelica Cueto

Publications and source records attributed to Maria Angelica Cueto.

18 recordsLinked to original sources

arXiv2026

Tropical methods for building real space sextics with totally real tritangent planes

This paper proposes the use of combinatorial techniques from tropical geometry to build the 120 tritangent planes to a given smooth algebraic space sextic. Although the tropical count is infinite, tropical tritangents come in 15 equivalence classes, each containing the tropicalization of exactly eight classical tritangents. Under mild genericity conditions on the tropical side, we show that liftings of tropical tritangents are defined over quadratic extensions of the ground field over which the input sextic curve is defined. When the input curve is real, we prove that every complex liftable member of a given tropical tritangent class either completely lifts to the reals or none of its liftings are defined over the reals. As our main application we use these methods to build examples of real space sextics with 64 and 120 totally real tritangents, respectively. The paper concludes with a discussion of our results in the arithmetic setting.

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arXiv2026

A lifting partition theorem for tropical tritangent classes to smooth space sextic curves

The set of tritangent planes to smooth tropical space sextic curves has 15 connected components, recording continuous displacements of planes preserving the tritangency condition. These 15 tritangent classes are polyhedral complexes in $\mathbf{R}^3$, and each of them contains the tropicalization of precisely eight tritangent planes to any smooth space sextic curve with the given tropicalization. Prior joint work of the authors with Len confirms that each tropical tritangent plane has 0, 1, 2, 4 or 8 lifts to classical tritangent planes defined over the algebraic closure of the field over which the original algebraic curve is defined. Our main theorem states that when the input classical curve is generic, then only six out of the ten possible partitions of 8 into powers of 2 arise from lifting multiplicities of tritangent classes. Furthermore, we show that these partitions are completely determined by the dimension of a suitable connected subcomplex of the class and the existence of a member with a tropical tangency of a predetermined combinatorial type.

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arXiv2023

Local tropicalizations of splice type surface singularities

Splice type surface singularities were introduced by Neumann and Wahl as a generalization of the class of Pham-Brieskorn-Hamm complete intersections of dimension two. Their construction depends on a weighted tree called a splice diagram. In this paper, we study these singularities from the tropical viewpoint. We characterize their local tropicalizations as the cones over the appropriately embedded associated splice diagrams. As a corollary, we reprove some of Neumann and Wahl's earlier results on these singularities by purely tropical methods, and show that splice type surface singularities are Newton non-degenerate complete intersections in the sense of Khovanskii. We also confirm that under suitable coprimality conditions on its weights, the diagram can be uniquely recovered from the local tropicalization. As a corollary of the Newton non-degeneracy property, we obtain an alternative proof of a recent theorem of de Felipe, González Pérez and Mourtada, stating that embedded resolutions of any plane curve singularity can be achieved by a single toric morphism, after re-embedding the ambient smooth surface germ in a higher-dimensional smooth space. The paper ends with an appendix by Jonathan Wahl, proving a criterion of regularity of a sequence in a ring of convergent power series, given the regularity of an associated sequence of initial forms.

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arXiv2023

Combinatorics and real lifts of bitangents to tropical quartic curves

Smooth algebraic plane quartics over algebraically closed fields have 28 bitangent lines. Their tropical counterparts often have infinitely many bitangents. They are grouped into seven equivalence classes, one for each linear system associated to an effective tropical theta characteristic on the tropical quartic. We show such classes determine tropically convex sets and provide a complete combinatorial classification of such objects into 41 types (up to symmetry). The occurrence of a given class is determined by both the combinatorial type and the metric structure of the input tropical plane quartic. We use this result to provide explicit sign-rules to obtain real lifts for each tropical bitangent class, and confirm that each one has either zero or exactly four real lifts, as previously conjectured by Len and the second author. Furthermore, such real lifts are always totally-real.

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arXiv2022

The Milnor fiber conjecture of Neumann and Wahl, and an overview of its proof

Splice type surface singularities, introduced in 2002 by Neumann and Wahl, provide all examples known so far of integral homology spheres which appear as links of complex isolated complete intersections of dimension two. They are determined, up to a form of equisingularity, by decorated trees called splice diagrams. In 2005, Neumann and Wahl formulated their Milnor fiber conjecture, stating that any choice of an internal edge of a splice diagram determines a special kind of decomposition into pieces of the Milnor fibers of the associated singularities. These pieces are constructed from the Milnor fibers of the splice type singularities determined by the subdiagrams on both sides of the chosen edge. In this paper we give an overview of this conjecture and a detailed outline of its proof, based on techniques from tropical geometry and log geometry in the sense of Fontaine and Illusie. The crucial log geometric ingredient is the operation of rounding of a complex logarithmic space introduced in 1999 by Kato and Nakayama. It is a functorial generalization of the operation of real oriented blowup. The use of the latter to study Milnor fibrations was pioneered by A'Campo in 1975.

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arXiv2019

Anticanonical tropical cubic del Pezzos contain exactly 27 lines

The classical statement of Cayley-Salmon that there are 27 lines on every smooth cubic surface in P^3 fails to hold under tropicalization: a tropical cubic surface in TP^3 often contains infinitely many tropical lines. Under mild genericity assumptions, we show that when embedded using the Eckardt triangles in the anticanonical system, tropical cubic del Pezzo surfaces contain exactly 27 tropical lines. In the non-generic case, which we identify explicitly, we find up to 27 extra lines, no multiple of which lifts to a curve on the cubic surface. We realize the moduli space of stable anticanonical tropical cubics as a four-dimensional fan in R^40 with an action of the Weyl group W(E_6). In the absence of Eckardt points, we show the combinatorial types of these tropical surfaces are determined by the boundary arrangement of 27 metric trees corresponding to the tropicalization of the classical 27 lines on the smooth algebraic cubic surfaces. Tropical convexity and the combinatorics of the root system E_6 play a central role in our analysis.

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arXiv2018

Tropical geometry of genus two curves

We exploit three classical characterizations of smooth genus two curves to study their tropical and analytic counterparts. First, we provide a combinatorial rule to determine the dual graph of each algebraic curve and the metric structure on the associated minimal Berkovich skeleton. Our main tool is the description of genus two curves via hyperelliptic covers of P^1 with six branch points. Given the valuations of these six points and their differences, our algorithm provides an explicit harmonic 2-to-1 map to a metric tree on six leaves. Second, we use tropical modifications to produce a faithful tropicalization in dimension three starting from a planar hyperelliptic embedding. Finally, we consider the moduli space of abstract genus two tropical curves and translate the classical Igusa invariants characterizing isomorphism classes of genus two algebraic curves into the tropical realm. While these tropical Igusa functions do not yield coordinates in the tropical moduli space, we propose an alternative set of invariants that provides new length data.

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arXiv2014

Cremona linearizations of some classical varieties

In this paper we present an effective method for linearizing rational varieties of codimension at least two under Cremona transformations, starting from a given parametrization. Using these linearizing Cremonas, we simplify the equations of secant and tangential varieties of some classical examples, including Veronese, Segre and Grassmann varieties. We end the paper by treating the special case of the Segre embedding of the n-fold product of projective spaces, where cumulant Cremonas, arising from algebraic statistics, appear as specific cases of our general construction.

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arXiv2014

How to repair tropicalizations of plane curves using modifications

Tropical geometry is sensitive to embeddings of algebraic varieties inside toric varieties. The purpose of this paper is to advertise tropical modifications as a tool to locally repair bad embeddings of plane curves, allowing the re-embedded tropical curve to better reflect the geometry of the input curve. Our approach is based on the close connection between analytic curves (in the sense of Berkovich) and tropical curves. We investigate the effect of these tropical modifications on the tropicalization map defined on the analytification of the given curve. Our study is motivated by the case of plane elliptic cubics, where good embeddings are characterized in terms of the j-invariant. Given a plane elliptic cubic whose tropicalization contains a cycle, we present an effective algorithm, based on non-Archimedean methods, to linearly re-embed the curve in dimension 4 so that its tropicalization reflects the j-invariant. We give an alternative elementary proof of this result by interpreting the initial terms of the A-discriminant of the defining equation as a local discriminant in the Newton subdivision.

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arXiv2014

Faithful tropicalization of the Grassmannian of planes

We show that the tropical projective Grassmannian of planes is homeomorphic to a closed subset of the analytic Grassmannian in Berkovich's sense by constructing a continuous section to the tropicalization map. Our main tool is an explicit description of the algebraic coordinate rings of the toric strata of the Grassmannian. We determine the fibers of the tropicalization map and compute the initial degenerations of all the toric strata. As a consequence, we prove that the tropical multiplicities of all points in the tropical projective Grassmannian are equal to one. Finally, we determine a piecewise linear structure on the image of our section that corresponds to the polyhedral structure on the tropical projective Grassmannian.

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arXiv2012

Implicitization of surfaces via geometric tropicalization

In this paper we further develop the theory of geometric tropicalization due to Hacking, Keel and Tevelev and we describe tropical methods for implicitization of surfaces. More precisely, we enrich this theory with a combinatorial formula for tropical multiplicities of regular points in arbitrary dimension and we prove a conjecture of Sturmfels and Tevelev regarding sufficient combinatorial conditions to compute tropical varieties via geometric tropicalization. Using these two results, we extend previous work of Sturmfels, Tevelev and Yu for tropical implicitization of generic surfaces, and we provide methods for approaching the non-generic cases.

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arXiv2011

Mixed Discriminants

The mixed discriminant of n Laurent polynomials in n variables is the irreducible polynomial in the coefficients which vanishes whenever two of the roots coincide. The Cayley trick expresses the mixed discriminant as an A-discriminant. We show that the degree of the mixed discriminant is a piecewise linear function in the Plucker coordinates of a mixed Grassmannian. An explicit degree formula is given for the case of plane curves.

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arXiv2011

Tropical secant graphs of monomial curves

The first secant variety of a projective monomial curve is a threefold with an action by a one-dimensional torus. Its tropicalization is a three-dimensional fan with a one-dimensional lineality space, so the tropical threefold is represented by a balanced graph. Our main result is an explicit construction of that graph. As a consequence, we obtain algorithms to effectively compute the multidegree and Chow polytope of an arbitrary projective monomial curve. This generalizes an earlier degree formula due to Ranestad. The combinatorics underlying our construction is rather delicate, and it is based on a refinement of the theory of geometric tropicalization due to Hacking, Keel and Tevelev.

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arXiv2011

Tropical mixtures of star tree metrics

We study tree metrics that can be realized as a mixture of two star tree metrics. We prove that the only trees admitting such a decomposition are the ones coming from a tree with at most one internal edge, and whose weight satisfies certain linear inequalities. We also characterize the fibers of the corresponding mixture map. In addition, we discuss the general framework of tropical secant varieties and we interpret our results within this setting. Finally, we show that the set of tree metric ranks of metrics on $n$ taxa is unbounded.

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arXiv2010

The maximum independent sets of de Bruijn graphs of diameter 3

The nodes of the de Bruijn graph B(d,3) consist of all strings of length 3, taken from an alphabet of size d, with edges between words which are distinct substrings of a word of length 4. We give an inductive characterization of the maximum independent sets of the de Bruijn graphs B(d,3) and for the de Bruijn graph of diameter three with loops removed, for arbitrary alphabet size. We derive a recurrence relation and an exponential generating function for their number. This recurrence allows us to construct exponentially many comma-free codes of length 3 with maximal cardinality.

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arXiv2010

An Implicitization Challenge for Binary Factor Analysis

We use tropical geometry to compute the multidegree and Newton polytope of the hypersurface of a statistical model with two hidden and four observed binary random variables, solving an open question stated by Drton, Sturmfels and Sullivant in "Lectures on Algebraic Statistics" (Problem 7.7). The model is obtained from the undirected graphical model of the complete bipartite graph $K_{2,4}$ by marginalizing two of the six binary random variables. We present algorithms for computing the Newton polytope of its defining equation by parallel walks along the polytope and its normal fan. In this way we compute vertices of the polytope. Finally, we also compute and certify its facets by studying tangent cones of the polytope at the symmetry classes vertices. The Newton polytope has 17214912 vertices in 44938 symmetry classes and 70646 facets in 246 symmetry classes.

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arXiv2009

Geometry of the restricted Boltzmann machine

The restricted Boltzmann machine is a graphical model for binary random variables. Based on a complete bipartite graph separating hidden and observed variables, it is the binary analog to the factor analysis model. We study this graphical model from the perspectives of algebraic statistics and tropical geometry, starting with the observation that its Zariski closure is a Hadamard power of the first secant variety of the Segre variety of projective lines. We derive a dimension formula for the tropicalized model, and we use it to show that the restricted Boltzmann machine is identifiable in many cases. Our methods include coding theory and geometry of linear threshold functions.

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arXiv2006

Some results on inhomogeneous discriminants

We study generalized Horn-Kapranov rational parametrizations of inhomogeneous sparse discriminants from both a theoretical and an algorithmic perspective. We show that all these parametrizations are birational and prove some results on the corresponding implicit equations. We also propose a combinatorial algorithm to compute the degree of inhomogeneous discriminantal surfaces associated to uniform matrices.

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