arXiv subjects

Mounir Nisse

Publications and source records attributed to Mounir Nisse.

At least 37 records · Page 2Linked to original sources

arXiv2026

Infinitesimal Variations of Hodge Structure for Singular Curves I

We study the infinitesimal variation of Hodge structure associated with families of reduced algebraic curves with singularities. The analysis applies to curves beyond the nodal case and is not restricted to plane curves, encompassing curves lying on smooth projective surfaces as well as families with more general isolated singularities. Using deformation-theoretic and residue-theoretic methods, we describe how the infinitesimal period map decomposes into local contributions supported at singular points, together with global constraints arising from the geometry of the normalization. While nodal singularities give rise to nontrivial rank-one contributions, other singularities may contribute only through higher-order local data or may be invisible at the infinitesimal level. As a consequence, we obtain sharp criteria for maximal infinitesimal variation in terms of numerical invariants of the curve, notably when the number of nodes satisfies the inequality $δ\ge g$, where $g$ denotes the genus of the normalization. We extend these results to curves on very general surfaces in projective three-space, showing that maximal variation persists on Picard-rank-one surfaces but fails for sufficiently large genus in the presence of higher ADE singularities. These results extend classical maximality phenomena in infinitesimal Hodge theory to a broader singular and geometric setting.

math.AG↗

arXiv2026

Residue Balancing on Singular Curves

This paper investigates residue maps and their spanning properties for singular algebraic curves, with particular emphasis on three interconnected themes: the \emph{scheme--theoretic residue span}, the \emph{residue--balancing principle}, and \emph{residue balancing in the presence of arbitrary singularities}. Starting from the theory of dualizing sheaves on nodal curves, we reinterpret canonical and higher--order differentials as meromorphic objects on the normalization whose local principal parts are constrained by explicit residue conditions. A key result is the scheme--theoretic residue span theorem, which asserts that % for nodal curves of geometric genus $g$ with $δ$ nodes, when $δ\ge g$ the residue functionals at the nodes span $H^0(C,ω_C)^\vee$, so canonical differentials are completely determined by their residue data. This provides a concrete, linear description of $H^0(C,ω_C)$ and yields powerful applications to deformation theory, Severi varieties, and moduli problems. \vspace{0.1cm} We then develop the residue--balancing principle, showing that global residue conditions on each irreducible component of a singular curve are equivalent to local balancing conditions at the singular points. This equivalence clarifies the local--to--global structure of dualizing sheaves and extends naturally to $k$--differentials. Finally, we address the case of arbitrary singularities, where nodes no longer suffice to describe local geometry. Using normalization and the conductor ideal, we formulate a refined balancing principle that replaces simple residue cancellation by higher--order and conductor--level constraints. Together, these results provide a unified framework for understanding how local singular behavior governs global differentials and their deformations.

math.AG↗

arXiv2026

Maximal Variation in the Moduli of Curves

We introduce and study the maximal-variation locus in families and moduli spaces of projective curves, defined via conductor-level balancing of meromorphic differentials on the normalization. This notion captures precisely when the space of canonical differentials behaves with the expected dimension under degeneration. We prove semicontinuity and openness results showing that maximal variation is stable in flat families, identify a natural determinantal degeneracy locus where maximal variation fails, and establish that this failure is governed entirely by the presence of non-Gorenstein singularities. In particular, all smooth and nodal curves satisfy maximal variation, while every non-Gorenstein singularity contributes explicitly and additively to degeneracy. We compute the expected codimension of degeneracy loci, describe their closure and adjacency relations in moduli, and explain how non-Gorenstein defects give rise to additional Hodge-theoretic phenomena in degenerations. This framework provides a uniform, intrinsic, and deformation-theoretically meaningful classification of degeneracy in spaces of canonical differentials.

math.AG↗

arXiv2025

On infinitesimal deformations of singular varieties I

The deformation theory of singular varieties plays a central role in understanding the geometry and moduli of algebraic varieties. For a variety $X$ with possibly singular points, the space of first-order infinitesimal deformations is given by $ T^1_X = \operatorname{Ext}^1_{\mathcal{O}_X}(Ω_X, \mathcal{O}_X), $ which measures the Zariski tangent space to the deformation functor of $X$. When $T^1_X = 0$, the variety is said to be \emph{rigid}; otherwise, nonzero elements of $T^1_X$ correspond to nontrivial first-order deformations. We investigate the structure of $T^1_X$ for singular varieties and provide cohomological and geometric criteria ensuring non-rigidity. In particular, we show that if the sheaf of tangent fields $T_X$ possesses nonvanishing cohomology $H^1(X, T_X)$ or if the local contributions $\mathcal{E}xt^1(Ω_X, \mathcal{O}_X)$ are supported on a positive-dimensional singular locus, then $T^1_X \neq 0$. For hypersurface singularities $X = \{ f = 0 \} \subset \mathbb{C}^{n+1}$, we recover the Jacobian criterion, \[ T^1_X \cong \frac{\mathbb{C}[x_0, \dots, x_n]}{(f, \partial f / \partial x_0, \dots, \partial f / \partial x_n)}, \] where the positivity of the Tjurina number $τ(X)$ characterizes the existence of nontrivial deformations. Moreover, non-rigidity arises when $X$ % appears as a cone over a projectively nonrigid variety. These criteria provide effective tools for detecting non-rigidity in both local and global settings, linking the vanishing of Ext and cohomology groups to the deformation behavior of singularities. The results contribute to a deeper understanding of the interplay between singularity theory, moduli, and the rigidity properties of algebraic varieties.

math.AG↗

arXiv2025

On infinitesimal deformations of singular varieties II

The deformation theory of affine cones over polarized projective varieties, initiated by Pinkham and further developed by Schlessinger and Wahl, is central to the study of singularities and graded deformation functors. For a projective variety $Y$ with ample line bundle $\mathcal L$, the affine cone $C(Y)$ carries a natural $\mathbb Z$-grading, and Pinkham's classical result identifies the graded pieces of its first-order deformation space: \[ T^{1}(C(Y))_m \,\cong\, H^{1}\!\left(Y,\,T_Y \otimes \mathcal L^{\otimes m}\right). \] This expresses that deformations of $C(Y)$ come from weighted deformations of $(Y,\mathcal L)$, with negative weights corresponding to smoothings and nonnegative weights to embedded deformations. % We give a streamlined proof of this isomorphism for possibly singular $Y$, using reflexive differentials and $\mathbb G_m$-equivariant deformations of the punctured cone. We then compute graded deformation spaces for several examples, illustrating phenomena from smoothability to rigidity. As an application, we obtain a practical rigidity criterion: if $ H^1(Y,\,T_Y \otimes \mathcal L^{\otimes m})=0 \text{ for all } m\in\mathbb Z, $ then $C(Y)$ is rigid. We exhibit explicit polarized varieties satisfying these vanishings, producing new rigid affine cones.

math.AG↗

arXiv2025

On infinitesimal deformations of singular varieties III

We study the affine cone over a reducible nodal curve $X$ obtained by gluing three projective lines along three pairs of points to form a connected curve of arithmetic genus $1$. We endow $X$ with a line bundle $L$ of multidegree $(4,3,3)$, and we show that $L$ is very ample, giving an embedding into $ \mathbb{P}^9$. We then analyze in detail the affine cone $ C(X) $ and determine its singular locus, which consists of three singular lines meeting at the vertex.

math.AG↗

arXiv2024

Links represented by phases of algebraic curves

The prime motivation behind this paper is to prove that any torus link can be realized as the union of the one-dimensional connected components of the set of critical values of the argument map restricted to a complex algebraic plane curve. Moreover, we give an explicit relation between the Newton polygon of such plane curves, and the number of components of the given torus link. This work aims to represent the starting point for a connection between knot theory, tropical geometry, and (co)amoebas.

math.AG↗

arXiv2024

Phase limit sets of linear spaces and discriminants

We show that the closure of the coamoeba of a linear space/hyperplane complement is the union of products of coamoebas of hyperplane complements coming from flags of flats, and relate this to the Bergman fan. Using the Horn-Kapranov parameterization of a reduced discriminant, this gives a partial description of the phase limit sets of discriminants and duals of toric varieties. When d=3, we show that each 3-dimensional component of the phase limit set of the discriminant is a prism over a discriminant coamoeba in dimension 2, which has a polyhedral description by a result of Nilsson and Passare.

math.AG↗

arXiv2024

Black hole thermodynamic free energy as $A$-discriminants

We show that the free energy $\mathcal{F}$ and temperature $T$ of black holes, considered as a thermodynamic system, can be viewed as an $A$-discriminant of an appropriately-defined polynomial. As such, mathematical results about $A$-discriminants may lead to implications about black hole thermodynamics. In particular, for static spacetimes with spherical, planar, or hyperbolic symmetry, the number of distinct thermodynamic phases depend on the number of distinct terms in the metric component $g_{tt}$. We prove that if $g_{tt}$ consists of $N_f$ distinct terms, then the $\mathcal{F}$-$T$ curve consists of $N_f-2$ cusps, which in turn leads to $N_f-1$ distinct thermodynamic phases. This result is applied to explicit examples of the Schwarzschild-AdS, Reissner--Nordström, power-law Maxwell, and Euler--Heisenberg black holes.

gr-qc↗

arXiv2024

Observing Algebraic Variety of Lee-Yang Zeros in Asymmetrical Systems via a Quantum Probe

Lee-Yang (LY) zeros, points on the complex plane of physical parameters where the partition function goes to zero, have found diverse applications across multiple disciplines like statistical physics, protein folding, percolation, complex networks etc. However, experimental extraction of the complete set of LY zeros for general asymmetrical classical systems remains a crucial challenge to put those applications into practice. Here, we propose a qubit-based method to simulate an asymmetrical classical Ising system, enabling the exploration of LY zeros at arbitrary values of physical parameters like temperature, internal couplings etc. Without assuming system symmetry, the full set of LY zeros forms an algebraic variety in a higher-dimensional complex plane. To determine this variety, we pro ject it into sets representing magnitudes (amoeba ) and phases (coamoeba ) of LY zeros. Our approach uses a probe qubit to initialize the system and to extract LY zeros without assuming any control over the system qubits. This is particularly important as controlling system qubits can get intractable with the increasing complexity of the system. Initializing the system at an amoeba point, coamoeba points are sampled by measuring probe qubit dynamics. Iterative sampling yields the entire algebraic variety. Experimental demonstration of the protocol is achieved through a three-qubit NMR register. This work expands the horizon of quantum simulation to domains where identifying LY zeros in general classical systems is pivotal. Moreover, by extracting abstract mathematical objects like amoeba and coamoeba for a given polynomial, our study integrates pure mathematical concepts into the realm of quantum simulations.

cond-mat.stat-mech↗

arXiv2021

Light-ring pairs from $A$-discriminantal varieties

When geodesic equations are formulated in terms of an effective potential $U$, circular orbits are characterised by $U=\partial_a U=0$. In this paper we consider the case where $U$ is an algebraic function. Then the condition for circular orbits defines an $A$-discriminantal variety. A theorem by Rojas and Rusek, suitably interpreted in the context of effective potentials, gives a precise criteria for certain types of spacetimes to contain at most two branches of light rings (null circular orbits), where one is stable and the other one unstable. We identify a few classes of static, spherically-symmetric spacetimes for which these two branches occur and show that the spacetimes with non-degenerate horizons do not have stable light rings.

gr-qc↗

arXiv2021

Describing Amoebas

An amoeba is the image of a subvariety of an algebraic torus under the logarithmic moment map. We consider some qualitative aspects of amoebas, establishing some results and posing problems for further study. These problems include determining the dimension of an amoeba, describing an amoeba as a semi-algebraic set, and identifying varieties whose amoebas are a finite intersection of amoebas of hypersurfaces. We show that an amoeba that is not of full dimension is not such a finite intersection if its variety is nondegenerate and we describe amoebas of lines as explicit semi-algebraic sets.

math.AG↗

arXiv2017

New Subexponential Fewnomial Hypersurface Bounds

Suppose $c_1,\ldots,c_{n+k}$ are real numbers, $\{a_1,\ldots,a_{n+k}\}\!\subset\!\mathbb{R}^n$ is a set of points not all lying in the same affine hyperplane, $y\!\in\!\mathbb{R}^n$, $a_j\cdot y$ denotes the standard real inner product of $a_j$ and $y$, and we set $g(y)\!:=\!\sum^{n+k}_{j=1} c_j e^{a_j\cdot y}$. We prove that, for generic $c_j$, the number of connected components of the real zero set of $g$ is $O\!\left(n^2+\sqrt{2}^{k^2}(n+2)^{k-2}\right)$. The best previous upper bounds, when restricted to the special case $k\!=\!3$ and counting just the non-compact components, were already exponential in $n$.

math.AG↗

arXiv2017

Amoeba-shaped polyhedral complex of an algebraic hypersurface

Given a complex algebraic hypersurface~$H$, we introduce a polyhedral complex which is a subset of the Newton polytope of the defining polynomial for~$H$ and enjoys the key topological and combinatorial properties of the amoeba of~$H.$ We provide an explicit formula for this polyhedral complex in the case when the spine of the amoeba is dual to a triangulation of the Newton polytope of the defining polynomial. In particular, this yields a description of the polyhedral complex when the hypersurface is optimal.

math.AG↗

arXiv2017

Metric Estimates and Membership Complexity for Archimedean Amoebae and Tropical Hypersurfaces

Given any complex Laurent polynomial $f$, $\mathrm{Amoeba}(f)$ is the image of its complex zero set under the coordinate-wise log absolute value map. We give an efficiently constructible polyhedral approximation, $\mathrm{ArchtTrop}(f)$, of $\mathrm{Amoeba}(f)$, and derive explicit upper and lower bounds, solely as a function of the number of monomial terms of $f$, for the Hausdorff distance between these two sets. We also show that deciding whether a given point lies in $\mathrm{ArchTrop}(f)$ is doable in polynomial-time, for any fixed dimension, unlike the corresponding problem for $\mathrm{Amoeba}(f)$, which is $\mathbf{NP}$-hard already in one variable. $\mathrm{ArchTrop}(f)$ can thus serve as a canonical low order approximation to start any higher order iterative polynomial system solving algorithm, such as homotopy continuation. $\mathrm{ArchTrop}(f)$ also provides an Archimedean analogue of Kapranov's Non-Archimedean Amoeba Theorem and a higher-dimensional extension of earlier estimates of Mikhalkin and Ostrowski.

math.AG↗

arXiv2016

Amoebas and coamoebas of linear spaces

We give a complete description of amoebas and coamoebas of $k$-dimensional very affine linear spaces in $(\mathbb{C}^*)^{n}$. This include an upper bound of their dimension, and we show that if a $k$-dimensional very affine linear space in $(\mathbb{C}^*)^{n}$ is generic, then the dimension of its (co)amoeba is equal to $\min \{ 2k, n\}$. Moreover, we prove that the volume of its coamoeba is equal to $π^{2k}$. In addition, if the space is generic and real, then the volume of its amoeba is equal to $\frac{π^{2k}}{2^k}$.

math.AG↗

arXiv2016

Geometry and a natural symplectic structure of phase tropical hypersurfaces

First, we define phase tropical hypersurfaces in terms of a degeneration data of smooth complex algebraic hypersurfaces in $(\mathbb{C}^*)^n$. Next, we prove that complex hyperplanes are diffeomorphic to their degeneration called phase tropical hyperplanes. More generally, using Mikhalkin's decomposition into pairs-of-pants of smooth algebraic hypersurfaces, we show that phase tropical hypersurfaces with smooth tropicalization, possess naturally a smooth differentiable structure. Moreover, we prove that phase tropical hypersurfaces possess a natural symplectic structure.

math.AG↗

arXiv2015

Higher convexity for complements of tropical varieties

We consider Gromov's homological higher convexity for complements of tropical varieties, establishing it for complements of tropical hypersurfaces and curves, and for nonarchimedean amoebas of varieties that are complete intersections over the field of complex Puiseaux series. Based on these results, we conjecture that the complement of a tropical variety has this higher convexity, and we prove a weak form of our conjecture for the nonarchimedean amoeba of a variety over the complex Puiseaux field. One of our main tools is Jonsson's limit theorem for tropical varieties.

math.AG↗