2.3 Tropical hypersurfaces

Tropical hypersurfaces are fascinating geometric objects that arise from tropical polynomials. They provide a bridge between algebra and geometry, offering insights into the structure of classical algebraic varieties through a combinatorial lens.

These piecewise linear objects are defined as the set of points where a tropical polynomial attains its maximum. By studying their properties, we can uncover deep connections between tropical geometry and classical algebraic geometry, shedding light on complex mathematical relationships.

Definition of tropical hypersurfaces

• A tropical hypersurface is the set of points in tropical projective space where a tropical polynomial attains its maximum
• Tropical hypersurfaces are piecewise linear objects that arise as limits of classical algebraic varieties over fields with valuation
• Studying tropical hypersurfaces provides insights into the combinatorial structure of algebraic varieties and their degenerations

Tropical polynomials

• Tropical polynomials are polynomials where the usual arithmetic operations are replaced by tropical operations: addition is replaced by maximum and multiplication is replaced by usual addition
• The coefficients of tropical polynomials are elements of the tropical semiring, which is the real numbers together with negative infinity
• Tropical polynomials can be used to define tropical hypersurfaces and study their geometric and combinatorial properties

Support of tropical polynomials

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• The support of a tropical polynomial is the set of exponent vectors of the monomials appearing in the polynomial with a coefficient not equal to negative infinity
• The convex hull of the support is the Newton polytope of the tropical polynomial
• The support of a tropical polynomial determines the combinatorial type of the corresponding tropical hypersurface

Coefficients in tropical polynomials

• The coefficients in a tropical polynomial determine the position of the tropical hypersurface in the tropical projective space
• Changing the coefficients of a tropical polynomial can lead to different combinatorial types of the corresponding tropical hypersurface
• The coefficients of a tropical polynomial can be used to study the moduli space of tropical hypersurfaces of a given degree

Tropical hypersurface as corner locus

• A tropical hypersurface can be viewed as the corner locus of a convex piecewise-linear function, i.e., the set of points where the function is not differentiable
• This interpretation allows for a geometric understanding of tropical hypersurfaces and their properties
• The corner locus perspective is useful for studying the relationship between tropical hypersurfaces and their dual subdivisions

Convex piecewise-linear functions

• A convex piecewise-linear function is a function that is the maximum of a finite number of affine linear functions
• The domains of linearity of a convex piecewise-linear function form a polyhedral subdivision of the space
• Convex piecewise-linear functions are closely related to tropical polynomials and can be used to define tropical hypersurfaces

Tropical hypersurface as non-differentiable locus

• The tropical hypersurface associated with a tropical polynomial $f$ is the set of points where the convex piecewise-linear function $\max\{a_i + \langle x, i \rangle\}$ is not differentiable, where $a_i$ are the coefficients of $f$ and $i$ runs over the support of $f$
• The non-differentiability of the convex piecewise-linear function occurs along the edges and vertices of the polyhedral subdivision induced by the function
• This characterization of tropical hypersurfaces allows for a geometric and combinatorial study of their properties

Newton subdivision of tropical hypersurfaces

• The Newton subdivision of a tropical hypersurface is a polyhedral subdivision of the Newton polytope of the associated tropical polynomial
• Each cell in the Newton subdivision corresponds to a domain of linearity of the convex piecewise-linear function defining the tropical hypersurface
• The Newton subdivision encodes important combinatorial information about the tropical hypersurface, such as its intersection with other hypersurfaces

Definition of Newton subdivision

• The Newton subdivision of a tropical polynomial $f$ is the polyhedral subdivision of the Newton polytope of $f$ induced by the convex piecewise-linear function $\max\{a_i + \langle x, i \rangle\}$, where $a_i$ are the coefficients of $f$ and $i$ runs over the support of $f$
• The cells of the Newton subdivision are the projections of the domains of linearity of the convex piecewise-linear function onto the Newton polytope
• The Newton subdivision can be computed using the regular subdivision induced by the coefficients of the tropical polynomial

Cells in Newton subdivision

• The cells in the Newton subdivision of a tropical hypersurface correspond to the components of the complement of the hypersurface in the tropical projective space
• The dimension of a cell in the Newton subdivision is equal to the codimension of the corresponding component of the complement of the tropical hypersurface
• The cells in the Newton subdivision can be used to study the topology and combinatorics of the tropical hypersurface

Dual graph of Newton subdivision

• The dual graph of the Newton subdivision of a tropical hypersurface is a graph where each vertex corresponds to a cell in the subdivision and two vertices are connected by an edge if the corresponding cells share a facet
• The dual graph encodes the adjacency relations between the components of the complement of the tropical hypersurface
• The dual graph can be used to study the connectivity and other properties of the tropical hypersurface

Balancing condition

• The balancing condition is a local condition on the coefficients of a tropical polynomial that ensures the smoothness of the corresponding tropical hypersurface
• It states that for each facet of the Newton subdivision, the sum of the lattice lengths of the edges adjacent to the facet, weighted by their corresponding coefficients, vanishes
• The balancing condition is a tropical analogue of the smoothness condition for algebraic varieties

Statement of balancing condition

• Let $f$ be a tropical polynomial and let $\tau$ be a facet of the Newton subdivision of $f$. For each edge $e$ adjacent to $\tau$, let $v_e$ be the primitive integer vector along $e$ pointing away from $\tau$ and let $a_e$ be the coefficient of the monomial corresponding to the endpoint of $e$ opposite to $\tau$. The balancing condition states that $\sum_e a_e v_e = 0$
• The balancing condition can be checked locally at each facet of the Newton subdivision
• If the balancing condition is satisfied, the tropical hypersurface is called smooth or balanced

Balancing condition vs smoothness

• The balancing condition is a necessary but not sufficient condition for the smoothness of a tropical hypersurface
• A tropical hypersurface is smooth if and only if it is locally the graph of a convex piecewise-linear function and the balancing condition is satisfied at each facet of the Newton subdivision
• Smoothness of tropical hypersurfaces is important for studying their intersection theory and relationship with classical algebraic geometry

Tropical Bézout's theorem

• Tropical Bézout's theorem is a fundamental result in tropical geometry that relates the intersection of tropical hypersurfaces to the mixed volume of their Newton polytopes
• It states that the number of stable intersection points of $n$ tropical hypersurfaces in $n$-dimensional tropical projective space, counted with multiplicities, is equal to the mixed volume of their Newton polytopes
• Tropical Bézout's theorem is a powerful tool for studying the intersection theory of tropical varieties and their relationship with classical algebraic geometry

Classical Bézout's theorem

• Classical Bézout's theorem states that the number of intersection points of $n$ algebraic hypersurfaces in $n$-dimensional projective space, counted with multiplicities, is equal to the product of their degrees
• It is a fundamental result in classical algebraic geometry and has numerous applications in enumerative geometry and other areas
• Tropical Bézout's theorem can be seen as a generalization of classical Bézout's theorem to the tropical setting

Statement of tropical Bézout's theorem

• Let $f_1, \ldots, f_n$ be tropical polynomials defining tropical hypersurfaces in $n$-dimensional tropical projective space. The number of stable intersection points of these hypersurfaces, counted with multiplicities, is equal to the mixed volume $\text{MV}(P_1, \ldots, P_n)$ of their Newton polytopes $P_1, \ldots, P_n$
• The mixed volume is a combinatorial invariant of the Newton polytopes that generalizes the notion of volume to the multivariate setting
• Tropical Bézout's theorem holds for any choice of coefficients of the tropical polynomials, as long as the intersection is stable

Intersection multiplicity in tropical Bézout's theorem

• The intersection multiplicity of a stable intersection point of tropical hypersurfaces is a local invariant that reflects the combinatorial structure of the intersection
• It can be computed using the balancing condition and the lattice lengths of the edges adjacent to the intersection point in the Newton subdivisions of the hypersurfaces
• The sum of the intersection multiplicities over all stable intersection points is equal to the mixed volume of the Newton polytopes, as stated in tropical Bézout's theorem

Stable intersection of tropical hypersurfaces

• A stable intersection of tropical hypersurfaces is an intersection that is preserved under small perturbations of the coefficients of the defining tropical polynomials
• Stable intersections are the tropical analogue of transversal intersections in classical algebraic geometry
• Studying stable intersections is crucial for developing a well-defined intersection theory for tropical varieties

Definition of stable intersection

• An intersection point of $n$ tropical hypersurfaces in $n$-dimensional tropical projective space is called stable if it is isolated and the intersection multiplicity remains constant under small perturbations of the coefficients of the defining tropical polynomials
• Equivalently, an intersection point is stable if the corresponding cells in the Newton subdivisions of the hypersurfaces intersect transversely and the balancing condition is satisfied at the intersection point
• Stable intersections are the key ingredient in the formulation of tropical Bézout's theorem and other results in tropical intersection theory

Stable intersection vs transversal intersection

• In classical algebraic geometry, a transversal intersection of hypersurfaces is an intersection where the tangent spaces of the hypersurfaces at the intersection point span the ambient space
• Stable intersections in tropical geometry can be seen as a combinatorial analogue of transversal intersections, where the role of tangent spaces is played by the cells in the Newton subdivisions
• While transversal intersections are generic in classical algebraic geometry, stable intersections are not necessarily generic in tropical geometry and may require specific choices of coefficients

Examples of tropical hypersurfaces

• Tropical hypersurfaces arise in various contexts and have interesting geometric and combinatorial properties
• Studying examples of tropical hypersurfaces helps develop intuition and showcases the rich structure of tropical varieties
• Some notable examples include linear tropical hypersurfaces, quadratic tropical hypersurfaces, and tropical hypersurfaces in higher dimensions

Linear tropical hypersurfaces

• A linear tropical hypersurface is the tropical vanishing locus of a tropical linear polynomial, i.e., a polynomial of the form $\max\{a_1 + x_1, \ldots, a_n + x_n\}$
• Linear tropical hypersurfaces are tropical hyperplanes and divide the tropical projective space into $n+1$ regions corresponding to the domains of linearity of the tropical polynomial
• The combinatorial structure of a linear tropical hypersurface is determined by the arrangement of the coefficients $a_1, \ldots, a_n$

Quadratic tropical hypersurfaces

• A quadratic tropical hypersurface is the tropical vanishing locus of a tropical quadratic polynomial, i.e., a polynomial of the form $\max\{a_{ij} + x_i + x_j\}$, where the maximum is taken over a subset of pairs $(i,j)$
• Quadratic tropical hypersurfaces are also known as tropical conics and have a rich combinatorial structure determined by the coefficients $a_{ij}$
• The Newton subdivision of a quadratic tropical hypersurface is a subdivision of the triangle with vertices $(0,0)$, $(2,0)$, and $(0,2)$, and the dual graph is a planar graph

Tropical hypersurfaces in higher dimensions

• Tropical hypersurfaces in higher dimensions are defined by tropical polynomials in more than two variables and exhibit a wide range of combinatorial and geometric phenomena
• The Newton subdivisions of higher-dimensional tropical hypersurfaces are subdivisions of higher-dimensional polytopes and can have intricate combinatorial structures
• Higher-dimensional tropical hypersurfaces arise in various applications, such as the study of amoebas of algebraic varieties, the tropicalization of moduli spaces, and the geometry of toric varieties