arXiv subjects
Publications and source records attributed to Mounir Nisse.
We give explicit real-algebraic equations for computing the singularities of amoeba contours, separating degeneracies of the logarithmic critical locus from coincidences of distinct critical lifts. For plane curves, this yields practical systems detecting nodes, cusps, and multiple branches. We also show that maximal sparsity does not maximize contour singularities, even for lattice triangles and parallelograms.
We establish new Newton-polygon bounds for the singularities of amoeba contours of smooth plane curves. Our cusp estimate refines the degree-four bound of Lang--Shapiro--Shustin, reducing its leading coefficient from $8$ to $4$ while retaining the normalized area, boundary lattice points, and directional widths of the Newton polygon. We also obtain a new multiplicity-sensitive bound for transverse $s$-nodes, with the natural decay factor $1/\binom{s}{2}$. The proofs combine logarithmic Gauss maps, normalized fiber products, ramification theory, and saturated off-diagonal intersection schemes, revealing the distinct geometric mechanisms governing cusps and multiple nodes.
We establish the first universal upper bounds for the real degree of the contour of the amoeba of a smooth complete intersection in the algebraic torus. Our approach extends the Pfaffian method of Lang--Shapiro--Shustin from hypersurfaces to arbitrary codimension by introducing a logarithmic conormal framework based on the logarithmic conormal bundle, the logarithmic Grassmann map, and determinantal rank conditions. We prove that, on suitable logarithmic conormal charts, the critical locus is locally defined by Schur--complement equations arising from the logarithmic conormal matrix. This yields explicit universal contour-degree estimates in both regimes $n\ge 2r$ and $n<2r$. We further replace total-degree arguments by Bernstein's theorem to obtain sparse mixed-volume bounds determined by the Newton polytopes of the transformed equations. These estimates are frequently much sharper than the corresponding universal bounds and provide a higher-codimensional analogue of the Lang--Shapiro--Shustin theory together with a new geometric interpretation of amoeba contours through logarithmic conormal geometry.
We derive new upper bounds for the real degree of the contour of the amoeba of algebraic curves and hypersurfaces. Our approach refines the Pfaffian method of Lang--Shapiro--Shustin by replacing large total-degree estimates with logarithmic-conormal elimination and sparse mixed-volume techniques based on the transformed Newton polytopes. This yields universal bounds together with significantly sharper sparse estimates for broad classes of Laurent polynomials. Several comparisons and explicit examples illustrate the improvement of the new bounds over the previously known universal estimates.
We study a converse problem to the theorem of Bergman and Bieri--Groves on logarithmic limit sets of algebraic subvarieties of the complex algebraic torus $(\mathbb C^*)^n$. We introduce the notion of \emph{finite logarithmic type}, a boundary condition formulated on toric compactifications in terms of coherent meromorphic extensions with uniformly bounded pole orders along toric boundary divisors. We prove that a closed reduced analytic subvariety of $(\mathbb C^*)^n$ whose logarithmic limit set is a finite rational spherical polyhedral complex of the expected dimension is algebraic whenever it is of finite logarithmic type. The proof relies on toric compactifications, coherent extension theory, Serre's GAGA theorem, and Chow's theorem. A principal result of the paper is the complete treatment of the one-dimensional case. We prove that every closed analytic curve in $(\mathbb C^*)^n$ with finite logarithmic limit set is algebraic. Consequently, the finiteness of the logarithmic limit set completely characterizes algebraicity for analytic curves, yielding a genuine converse to Bergman's theorem in dimension one. These results establish new links between logarithmic limit sets, tropical geometry, toric geometry, and complex analytic geometry.
We investigate the real algebraic complexity of contours of amoebas associated with algebraic hypersurfaces and complete intersections in complex algebraic tori. Motivated by the foundational estimates of Lang--Shapiro--Shustin \cite{LSS}, we develop a toric and logarithmic approach relating contour geometry to logarithmic Gauss maps, mixed volumes, and Newton polytope geometry. We prove that the logarithmic criticality equations admit substantially sharper asymptotic behavior than the classical Pfaffian bounds, reducing the expected growth from order $d^{2n}$ to order $d^n$. We further formulate a conjectural asymptotic theory for complete intersections based on logarithmic Grassmannian geometry and Schubert degeneracy loci. The work establishes new connections between amoeba theory, real algebraic geometry, tropical geometry, and logarithmic toric geometry.
We study tropical degree bounds, stable tropical intersections, and tropical Bézout-type estimates through the geometry of Newton polytopes, mixed subdivisions, and lattice indices. We establish an upper bound for the tropical degree of a tropical hypersurface in terms of the $\ell^1$-diameter of the support of its defining tropical polynomial. We then investigate stable tropical intersections using the intrinsic framework of Jensen and Yu and show that local stable intersection multiplicities admit explicit determinant and lattice-volume interpretations. For transverse tropical complete intersections, we recover tropical Bernstein-type formulas through fully mixed cells in mixed subdivisions of Newton polytopes. We further analyze the non-complete-intersection setting and prove that mixed volumes still provide natural local upper bounds after transverse local reductions. The results reveal a direct geometric relationship between tropical degrees, Newton polytopes, lattice covolumes, and stable tropical intersection theory.
The topology of amoebas of complex algebraic hypersurfaces is deeply connected to the combinatorics of the Newton polytope and the convex geometry of the Ronkin function. A long-standing conjecture of Passare and Rullgard asserts that the amoeba of a maximally sparse Laurent polynomial, whose support consists exactly of the vertices of its Newton polytope, must be solid, meaning that the complement of the amoeba has precisely as many connected components as the number of vertices of the Newton polytope. In this paper we prove this conjecture. The proof is based on a detailed analysis of the stability of the linearity domains of the Ronkin function under tropical degenerations of Laurent polynomials. We show that in the maximally sparse case no new slopes corresponding to interior lattice points can appear, forcing the amoeba complement to have the minimal possible topology. In addition, we establish stability results for the spines of degenerating amoebas, prove that the associated Newton subdivision stabilizes and coincides with the tropical subdivision for sufficiently small parameters, and derive geometric criteria controlling the appearance of lattice points in the dual subdivision. These results lead to a classification of three distinct regimes governing the topology of amoeba complements according to the position of the support relative to the Newton polytope.
We study the infinitesimal variation of Hodge structure for families of algebraic curves and extend the classical theory from smooth curves to singular and non--planar settings. Using the deformation space $\mathrm{Ext}^1(Ω_X,\mathcal O_X)$ and the dualizing sheaf, we define a singular analogue of maximal infinitesimal variation. For equisingular families of plane curves with planar Gorenstein singularities, we prove that the infinitesimal variation attains maximal rank equal to the arithmetic genus. We show that the rank decomposes into a geometric contribution from the normalization and a singular contribution measured by the $δ$--invariants. For non--equisingular degenerations, the rank defect equals the drop of the total $δ$--invariant and admits an interpretation in terms of vanishing cycles and mixed Hodge structures. We further extend the results to non--planar curves under suitable Petri and deformation conditions.
We investigate higher--order variation of Hodge structure for families of smooth hypersurfaces and complete intersections through the notion of $I$--maximal variation. Using Griffiths' description of primitive cohomology, we interpret the infinitesimal variation of Hodge structure and the $n$--fold Yukawa coupling as graded multiplication maps in the Jacobian ring. Our main result shows that the Strong Lefschetz property of the Jacobian ring provides the algebraic mechanism ensuring $I$--maximal variation. In particular, we prove that smooth hypersurfaces of degree $d\ge n+2$ and smooth complete intersections with $κ>0$ exhibit $I$--maximal variation. We further establish that for complete intersections of general type the infinitesimal Torelli property is equivalent to the nondegeneracy of the Yukawa coupling. Finally, we analyze degenerations and show that the failure of the Strong Lefschetz property leads to degeneration of the Yukawa coupling and the loss of $I$--maximal variation. These results identify the Lefschetz property of the Jacobian ring as the fundamental algebraic structure governing maximal variation of Hodge structure.
We study equisingular deformation problems for curves and surfaces in algebraic families, with particular emphasis on situations where nodal behavior is no longer generic. Extending classical Severi theory, we develop deformation--theoretic criteria ensuring the existence of deformations with isolated singularities of minimal type, including cusps on curves and ordinary double points on curves and surfaces in threefolds. Under unobstructedness and surjectivity assumptions for natural global--to--local maps of normal bundles, we prove maximality results showing that the number of such singularities is governed by the global realizability of equisingular deformation directions rather than by numerical invariants alone. Logarithmic semiregularity allows these results to persist in degenerations with normal crossings special fibers. We further explain how these singularities arise as boundary phenomena of equigeneric Severi strata and outline applications to refined Severi counts via logarithmic and tropical methods.
We study equisingular deformation problems for curves and surfaces in algebraic families, with particular emphasis on situations where nodal behavior is no longer generic. Extending classical Severi theory, we develop deformation--theoretic criteria ensuring the existence of deformations with isolated singularities of minimal type, including cusps on curves and ordinary double points on curves and surfaces in threefolds. Under unobstructedness and surjectivity assumptions for natural global--to--local maps of normal bundles, we prove maximality results showing that the number of such singularities is governed by the global realizability of equisingular deformation directions rather than by numerical invariants alone. Logarithmic semiregularity allows these results to persist in degenerations with normal crossings special fibers. We further explain how these singularities arise as boundary phenomena of equigeneric Severi strata and outline applications to refined Severi counts via logarithmic and tropical methods.
This paper develops a systematic approach to infinitesimal variations of Hodge structure for singular and equisingular families by means of logarithmic geometry and residue theory. The central idea is that logarithmic vector fields encode precisely those deformation directions that preserve singularities and act trivially on Hodge structures, while the effective variation is entirely governed by residue calculus. This viewpoint provides a conceptual reinterpretation of classical results of Griffiths, Green, and Voisin, and extends them to settings involving singular varieties and equisingular deformations. The resulting framework yields a geometric explanation for the appearance of Jacobian rings in infinitesimal Hodge theory and clarifies the structure of deformation spaces underlying Severi varieties and related moduli problems.
We study infinitesimal Torelli problems and infinitesimal variations of Hodge structure for families of curves arising in singular and extrinsically constrained geometric settings. Motivated by the Green--Voisin philosophy, we develop an explicit approach based on Poincaré residue calculus, allowing a uniform treatment of smooth, singular, and equisingular situations. In particular, we prove infinitesimal Torelli theorems for general equisingular plane curves of sufficiently high degree and construct relative IVHS exact sequences for curves lying on smooth projective threefolds. Our results show that maximal infinitesimal variation of Hodge structure persists even after imposing strong extrinsic conditions, such as fixed degree and prescribed singularities, and in the presence of isolated planar singularities. The methods presented here provide a concrete and geometric realization of Jacobian-type constructions and extend the Green--Voisin philosophy to singular and equisingular settings and provide a unified residue--theoretic framework for Torelli--type problems across dimensions and codimensions.
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.
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.
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.
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.