Quadratically Enriched Enumerative Geometry

• Quadratically enriched enumerative geometry is the extension of classic enumerative invariants to a framework using the Grothendieck–Witt ring and A¹-homotopy techniques.
• It combines tropical geometry with combinatorial local models to compute GW(k)-valued intersection numbers that capture both arithmetic and quadratic data.
• The approach unifies complex counts, signed real counts, and arithmetic invariants, advancing both theoretical insights and practical computations.

Quadratically enriched enumerative geometry refers to the extension of classical enumerative problems—originally formulated over algebraically closed fields and valued in ℤ—to a framework where enumerative invariants take values in the Grothendieck–Witt group GW(k) of nondegenerate symmetric bilinear forms over a base field k. Within tropical enumerative geometry, this enrichment is achieved by combining methodologies from tropical geometry and the machinery of $\mathbb{A}^1$-homotopy theory, resulting in GW(k)-valued tropical intersection invariants. This synthesis permits the computation and interpretation of intersection multiplicities and enumerative numbers not just over ℂ or ℝ, but over arbitrary fields, reflecting arithmetic and quadratic structure invisible to usual counts (Puentes et al., 2022).

1. The Grothendieck–Witt Ring and $\mathbb{A}^1$-Enumerative Geometry

The Grothendieck–Witt ring GW(k) is the group completion of isometry classes of nondegenerate symmetric bilinear forms on finite-dimensional k-vector spaces. It is generated by rank one classes ⟨a⟩ associated to $(x, y) \mapsto a x y$ for $a\in k^\times/(k^\times)^2$, with relations:

• ⟨a⟩⊗⟨b⟩ = ⟨ab⟩,
• ⟨a⟩⊕⟨b⟩ = ⟨a+b⟩⊕⟨ab(a+b)⟩, for $a,b,a+b\neq0$.

The hyperbolic form $h = \langle1\rangle + \langle-1\rangle$ satisfies $\langle a\rangle \oplus \langle -a\rangle = h$.

• The rank map GW(k) → ℤ recovers classical complex counts.
• For $k = \mathbb{R}$, the signature map GW(ℝ) → ℤ retrieves signed real counts (e.g., Welschinger invariants).

In $\mathbb{A}^1$-enumerative geometry, ordinary degrees and multiplicities at geometric solutions are replaced by $\mathbb{A}^1$-degrees taking values in GW(k). If $f : k^n \to k^n$ has an isolated, nondegenerate zero at $z$ with residue field $k(z)$, the local degree is

$$\deg_{k(z)}(f) = \mathrm{Tr}_{k(z)/k}\langle \det \, \mathrm{Jac}\, f(z)\rangle \in \mathrm{GW}(k).$$

This enriches enumerative geometry by encoding, at each geometric solution, the quadratic information of the local Jacobian.

2. Classical and Tropical Intersections: Transition to Quadratic Enrichment

Classical tropical geometry rephrases enumerative geometry problems as combinatorial counts of tropical (piecewise linear) hypersurfaces. Tropical intersection multiplicity at a transverse intersection point p of n tropical hypersurfaces $V_1, \dots, V_n \subset \mathbb{R}^n$ is defined as the lattice volume $m = |\det M|$ of the dual cell $P$ in the Newton polytope subdivision (Puentes et al., 2022). The global count over all intersection points reproduces mixed volume statements, leading to tropical versions of Bézout's and Bernstein–Kushnirenko theorems.

Quadratic enrichment is realized as follows:

• Given a Viro polynomial $f(t, x)=\sum_{I\in A} \alpha_I x^I t^{\varphi(I)}$ with $\alpha_I\in k^\times, \varphi$ convex, tropicalization yields a hypersurface $V$ in $\mathbb{R}^n$ and its dual subdivision. The “enrichment data” records $\alpha_v$ (modulo squares) at each vertex $v$.
• An enriched tropical hypersurface consists of $(V, \{\alpha_v\})$.

At a tropically transverse intersection point $p$ of enriched tropical hypersurfaces, the local “quadratically enriched” intersection number is

$$_p(V_1, ..., V_n) := \sum_{z: \mathrm{trop}(z)=p} \mathrm{Tr}_{k(z)/k}\langle \det \, \mathrm{Jac}(f_1, ..., f_n)(z) \rangle \in \mathrm{GW}(k)$$

with the rank of $_p$ giving the classical multiplicity $m$.

3. Main Theorems: Local and Global GW(k)-Valued Intersection Theory

The principal theorems in quadratically enriched tropical enumerative geometry include:

• Combinatorial Local Multiplicity Given enriched tropical hypersurfaces $V_1, ..., V_n$ meeting tropically transversely at $p$, with dual parallelepiped $P$ of volume $m$ and set of “odd” vertices $v_1, ..., v_q$ ($v_j \equiv (1,...,1)\mod 2$), if $\epsilon_P(v_\ell)=\pm1$ is the sign of the oriented minor and $\alpha_{v_\ell}$ is the enrichment coefficient at $v_\ell$, then

$$_p(V_1,\ldots,V_n) = \sum_{\ell=1}^q \epsilon_P(v_\ell)\, \alpha_{v_\ell} + \frac{m-q}{2}\cdot h$$

in GW(k).

• Enriched Tropical Bézout Theorem If $V_1,...,V_n$ have Newton polytopes $\Delta_{d_i}$ and $\sum d_i\equiv n+1 \bmod 2$, then

$\sum_{p} \, _p(V_1,...,V_n) = \frac{d_1 \cdots d_n}{2}\cdot h \quad \text{in } GW(k)$

giving the expected number of complex solutions as the rank and the Welschinger signed count as the signature.

• Enriched Bernstein–Kushnirenko Theorem For $f_i$ with Newton polytopes $\Delta_i$ and $\Delta_1 + \cdots + \Delta_n$ without odd boundary points, for a generic system $f_1 = \cdots = f_n = 0$ in $(k^\times)^n$,

$$\sum_{z}\mathrm{Tr}_{k(z)/k}\langle \det \, \mathrm{Jac}(z) \rangle = \frac{\mathrm{MVol}(\Delta_1,...,\Delta_n)}{2} h$$

with explicit correction terms in the non-orientable case (Puentes et al., 2022).

4. Combinatorial and Algebraic Methodology

Quadratically enriched invariants are computed by reducing to local models:

• Near $p$, each $V_i$ is modeled by a binomial $\alpha_i x^{I_i}+\beta_i x^{J_i} = 0$; the dual parallelepiped $P$'s corners $v_A$ are determined combinatorially.
• The local intersection number is expressed via traces and determinants as in the formula above.
• The combinatorial recipe utilizes, for each odd corner, its enrichment label, together with balancing signs from the oriented minors.

This approach yields an explicit combinatorial formula for local GW(k)-valued multiplicities. The odd-vertex labeling mechanism is crucial for capturing nontrivial quadratic structure present already in characteristic counts and sign assignments (e.g., Welschinger signs in real geometry).

5. Examples and Specializations

In dimension two, the intersection of two tropical curves of degrees $d_1, d_2$ generically consists of $d_1d_2$ points.

• If $d_1 + d_2 \equiv 1 \bmod 2$, the global quadratically enriched count is $\frac{d_1 d_2}{2} h$ in GW(k).
• In explicit graphical situations, local counts can be read off from labeling odd vertices and their coefficients as dictated by the dual subdivision and prescribed $\alpha$'s.

Special cases include:

• $k = \mathbb{C}$: $GW(\mathbb{C}) \cong \mathbb{Z}$ (by rank), recovering complex counts.
• $k = \mathbb{R}$: $GW(\mathbb{R}) \cong \mathbb{Z} \oplus \mathbb{Z}$ (rank, signature), recovering signed real counts (Welschinger-type).
• Toric ambient spaces, notably $(\mathbb{P}^1)^n$, $\mathbb{P}^1 \times \mathbb{P}^1$, and Hirzebruch surfaces, where combinatorial orientability is equivalent to the relative orientability of the vector bundle, and the mixed volume formulas in GW(k) reflect the arithmetic refinement.

6. Implications and Broader Context

Quadratically enriched tropical intersections unify and extend classical, real, and arithmetic enumerative invariants under a GW(k)-valued umbrella. This framework exposes arithmetic information such as discriminants, norms, and Witt residues, and encodes parity phenomena and subtleties of field of definition not visible in integer-valued counts (Puentes et al., 2022). The methods provide a combinatorial mechanism for computing $\mathbb{A}^1$-enumerative GW(k)-invariants for hypersurface intersections in toric spaces, cementing the role of tropical geometry as an effective tool for enriched and arithmetic enumerative geometry.

The rank and signature functors recover standard counts: for $k = \mathbb{C}$ the complex Gromov–Witten numbers, for $k = \mathbb{R}$ the signed Welschinger invariants. The approach is compatible with specialization and deformation, and leads to new arithmetic invariants over finite and $p$-adic fields, as well as to systematic GW(k)-refinements of classical intersection theorems.

7. Further Developments

Subsequent work has extended these ideas to higher dimensions and broader classes of varieties. The combinatorial machinery underpinning local and global computations of GW(k)-valued invariants continues to influence the formulation of arithmetic refinements in tropical correspondence theorems and the development of quadratic enhancements for enumerative invariants in both classical and tropical settings (Puentes et al., 2022). The direct combinatorial correspondence—via dual subdivisions, odd corners, and sign-labeled enrichments—serves as a foundational recipe for the computation of A¹-enumerative invariants across a spectrum of ambient geometries.