Incidence-Variety Compactification (IVC)

• Incidence-variety compactification (IVC) is a method that extends loci defined by incidence conditions in moduli spaces by encoding degenerations and boundary behaviors.
• It utilizes explicit constructions such as Cartier divisors and blow-up models to compute enumerative invariants and refine intersection theory.
• The approach tracks zero-breaking, spin/parity invariants, and combinatorial data, thereby enhancing calculations in moduli theory and algebraic geometry.

The incidence-variety compactification (IVC) is a foundational algebro-geometric technique for extending loci defined by geometric incidence conditions—such as the intersection of effective cycles, divisors with prescribed vanishing, or strata of meromorphic differentials—across partial or full compactifications of moduli spaces. The IVC systematically encodes degenerations and boundary behavior in families of algebraic cycles or differentials, often by realizing the incidence locus as a Cartier divisor or through explicit algebro-geometric models such as blow-ups along ideal sheaves. These compactifications play a central role in enumerative geometry, Hodge-theoretic intersection theory, and the compactification of strata of moduli spaces. Their development is closely linked with the construction of tautological bundles, the study of refined Gysin correspondences, and the calculation of enumerative invariants.

1. Foundations: The Incidence Locus and Cartier Divisors

Let $P$ be a smooth projective variety of dimension $n$ over an algebraically closed field. Denote by $C_a(P)$ the disjoint union of Chow varieties parameterizing effective $a$-cycles of all degrees, and let $b = n-a-1$. The product $C_a(P) \times C_b(P)$ carries the incidence locus $I$, defined as

$$I = \{ (A, B) \mid \mathrm{supp}(A) \cap \mathrm{supp}(B) \neq \emptyset \}.$$

This is a closed algebraic subset. A central result is that $I$, or more precisely its closure $\overline{I} = I$, is an effective Cartier divisor on $n$0 at least over the open locus of pairs with properly intersecting supports. The associated determinant line bundle is constructed on the product of Hilbert schemes of $n$1- and $n$2-dimensional subschemes via the derived pushforward of the tensor product of universal families. Explicitly, if $n$3 are the pulled-back structure sheaves of the universal families,

$n$4

defines the incidence line bundle on $n$5. The main descent theorem asserts that $n$6 canonically trivializes away from the incidence locus and descends to a line bundle $n$7 on the corresponding product of Chow varieties, whose canonical section vanishes exactly on $n$8; i.e., the incidence locus is globally cut out by a Cartier divisor descending from the determinant construction on the Hilbert scheme side (Ross, 2010).

2. Incidence-Variety Compactification in Moduli of Differentials

Within the context of strata of canonical, $n$9-canonical, or $C_a(P)$0-differentials over moduli spaces of curves, the IVC provides a precise, functorial compactification that respects the incidence conditions on divisor data across degenerations. Given integers $C_a(P)$1 summing to $C_a(P)$2, the ordered incidence variety

$C_a(P)$3

sits naturally inside the total space of the Hodge bundle over the moduli of $C_a(P)$4-pointed stable curves. The IVC is defined as the closure (modulo permutations of equal order zeros) of this locus in the compactified Hodge bundle, retaining explicit information about the locations and orders of zeros (and permitted poles) even in degenerations to nodal or reducible curves.

A central algebro-geometric tool is the plumbing cylinder construction, which enables the smoothing of nodal differentials in a way compatible with the assignment of zeros and poles across components (Gendron, 2015). Key features retained in the IVC include:

• Zero-breaking: The ability to track the splitting or migration of zeros across the rational (or exceptional) bridges introduced during degeneration.
• Incidence conditions on components: The incidence relation is enforced on each irreducible component, ensuring the sum of orders matches the divisor of the (limit) differential.

Extension of parity invariants (theta-characteristics and spin structure) across compact type degenerate curves is realized via associated line bundles on the resolution of the nodal curves, allowing precise distinction of topological and geometric features of boundary points that are obscured in other (e.g., Deligne–Mumford) compactifications.

3. Graph-Sum Formalism and Tautological Intersection Theory

The IVC admits explicit combinatorial and intersection-theoretic descriptions, notably in the moduli of $C_a(P)$5-differentials over curves with rational tails. Here, the incidence loci are determined by prescribed vanishing orders at the marked points, and their classes in the tautological ring are computed via decorated stable graphs. Each dual graph $C_a(P)$6 (with vertices, edges, and legs encoding the degeneration type) is decorated with psi-classes and equipped with explicit weightings reflecting the combinatorics of zeros colliding and distributing on the rational components.

For instance, for the incidence locus $C_a(P)$7 in the projectivized $C_a(P)$8-th Hodge bundle, the class of its closure is given as a sum

$C_a(P)$9

where all coefficients and decorations are explicit and combinatorially controlled (Gheorghita et al., 2021). This formalism enables recursive intersection computations, evaluation of tautological relations, and explicit calculations in low-dimensional or specialized loci (e.g., n=1,2 cases, one heavy point, Logan divisor).

4. Explicit Algebraic Models and Blow-Up Descriptions

In genus zero, an explicit algebro-geometric model for the IVC is constructed as the blow-up of the Deligne–Mumford–Knudsen compactification of the moduli space $a$0 along an ideal sheaf. For fixed $a$1 and integer vector $a$2 with $a$3 and $a$4, the relevant stratum of $a$5-differentials and its compactification admit the identification

$a$6

where $a$7 is a boundary-supported coherent ideal sheaf built from explicit local monomials depending on the dual graph of the boundary point. The exceptional divisor and the tautological line bundle have closed-form expressions in terms of explicit sums over the combinatorial data of the partitioning of markings and the pole/zero profile, yielding recursive volume formulas for the spaces of flat metrics with prescribed conical angles (Nguyen, 2021).

5. Applications: Enumerative Invariants, Height Pairings, and Moduli Theory

The IVC provides a robust geometric framework for computing and interpreting various invariants:

• Enumerative Geometry: The intersection class $a$8 allows evaluation of enumerative invariants, such as the number of incidences in families of cycles and degrees of corresponding divisors (Ross, 2010).
• Tautological Rings: The explicit graph-sum and decorated-graph formalism establishes relations and vanishing theorems within the tautological ring of the moduli space, with direct links to relations like the Logan divisor and other key algebraic cycles (Gheorghita et al., 2021).
• Volumes of Moduli Spaces: In genus zero, the compactification as a blow-up provides both a conceptual and computational method for volume calculations of spaces of differentials via intersection numbers of tautological divisors (Nguyen, 2021).
• Arakelov-Theoretic Height Pairings: The line bundle $a$9 and its metrization support natural height pairings over $b = n-a-1$0, extending the intersection-theoretic structure to arithmetic settings (Ross, 2010).

6. Distinctions, Special Features, and Boundary Behavior

A salient distinction of the IVC relative to other compactifications (e.g., Deligne–Mumford, stable pairs) is its retention of fine incidence and divisor-position data at the boundary. Notably:

• Zero-breaking and Boundary Excess: The IVC reflects all possible ways zeros can distribute on rational tails or exceptional components, thereby capturing higher-dimensional boundary strata and excess phenomena not visible in the stable curve plus line bundle perspective (Gendron, 2015).
• Spin/Parity Invariants: The compactification enables the extension of spin structures and associated invariants (theta-characteristics, Arf invariants) to the boundary, distinguishing components such as hyperelliptic versus odd minimal strata even in the presence of boundary degenerations (Gendron, 2015).
• Algebraic Tractability: The blow-up description supplies a manageable global algebro-geometric model, allowing explicit calculations and direct connections to the geometry of the moduli space (Nguyen, 2021).

7. Key Examples and Special Cases

• Classical Incidence Divisors: For $b = n-a-1$1, the IVC construction recovers the Grassmannian incidence divisor via the Hilbert–Chow morphism, with compatibility to ruled-join constructions and classical enumerative relations (Ross, 2010).
• Strata of $b = n-a-1$2-Canonical Differentials: The IVC provides explicit classes and relations for loci of $b = n-a-1$3-differentials satisfying vanishing and pole conditions, instrumental in studying the structure and Kodaira dimension of moduli spaces of abelian or quadratic differentials (Gendron, 2015).
• Genus Zero and Flat Metrics: The explicit blow-up model in genus zero supports the derivation of recursive formulae for the Masur–Veech volumes of strata, encompassing cases of quadratic differentials and general $b = n-a-1$4 (Nguyen, 2021).

The incidence-variety compactification thus furnishes a versatile and canonical geometric compactification for incidence-type moduli problems, retaining essential intersection-theoretic, incidence, and degeneration information required across several domains of algebraic geometry, Hodge theory, and arithmetic geometry.