Tyurin Degeneration in Algebraic Geometry

Updated 25 October 2025

Tyurin degeneration is a geometric process that splits a smooth algebraic variety into two components joined by a smooth anticanonical divisor.
It plays a pivotal role in mirror symmetry, enumerative geometry, and moduli space compactifications with explicit combinatorial and deformation theoretic models.
The approach employs toric and log deformation theories to compute invariants and bridges classical geometry with derived categorical perspectives.

A Tyurin degeneration refers to a precise geometric process in which a smooth algebraic variety—most notably a K3 surface or Calabi–Yau manifold—degenerates into a union of two quasi-Fano varieties or rational surfaces, glued transversely along a smooth anticanonical divisor (which is typically an elliptic curve or a K3 surface itself). This construction underlies key phenomena in algebraic geometry, mirror symmetry, enumerative geometry, and categorical invariants, and plays a central role in the paper of moduli spaces and wall structures in mirror symmetry. Below, the principal technical aspects and implications of Tyurin degenerations are systematically discussed.

1. Formal Definition and Core Structure

A Tyurin degeneration is generally defined as a flat family

$\pi : \mathcal{X} \to D$

where $D \subset \mathbb{C}$ is the unit disc, the general fiber $X_t$ ( $t \neq 0$ ) is smooth (e.g., a K3 surface, Calabi–Yau manifold), and the central fiber is

$\mathcal{X}_0 = X_1 \cup_Z X_2$

with $X_1, X_2$ quasi-Fano varieties or rational surfaces meeting transversely along a smooth anticanonical divisor $Z$ (i.e., $Z \in |{-K_{X_i}}|$ for each $i$ ). The normal bundle condition

$N_{Z/X_1} \otimes N_{Z/X_2} \simeq \mathcal{O}_Z$

is required to ensure smoothability of the union. For K3 surfaces, the central fiber consists of a pair of rational surfaces glued along a smooth elliptic curve, which must be anticanonical on both components (Giovenzana et al., 20 May 2024, Doran et al., 2023, Jones, 6 Feb 2025).

2. Toric Degenerations, Log Deformation Theory, and Explicit Models

A toric degeneration is a special instance of Tyurin degeneration where the central fiber is a union of toric varieties (e.g., projective spaces) meeting along toric strata. Such degenerations admit explicit combinatorial descriptions that facilitate the analysis of curves and disks on the variety. For example, in the degeneration of a K3 or Calabi–Yau hypersurface given locally by

$XY + tZ = 0$

the central fiber at $t=0$ decomposes into toric pieces (Nishinou, 2011). The paper of deformation theory in these settings is then reduced to local computations using logarithmic structures, with the obstruction theory accessible via log normal sheaves and Kuranishi maps. The computation of incidence conditions in the central fiber often directly determines enumerative invariants, such as the exact count of lines on the quintic Calabi–Yau

$575 \times 5 = 2875$

via combinatorial corrections (Nishinou, 2011). In higher dimensions, toric degenerations of Hirzebruch scrolls yield factorization of the anticanonical section into two pieces whose intersection is itself Calabi–Yau (Berglund et al., 2022).

3. Enumerative Geometry, Modularity, and Wall Structures

Tyurin degenerations serve as powerful tools in enumerative geometry, enabling explicit calculation of curve counts, disk counts, and enumerative invariants via combinatorial and local analytic approaches on the reducible central fiber. In K3-fibered Calabi–Yau threefolds, Tyurin degeneration facilitates the connection between Gromov–Witten (GW), Gopakumar–Vafa (GV), and Noether–Lefschetz (NL) invariants, and their modular properties. For instance, the generating functions for vertical GV and NL numbers in families with Tyurin degeneration exhibit transformation properties under congruence subgroups such as

$\Gamma_0(m)^+$

and modular behavior is encoded in modular curves parametrizing boundary divisors of the family (Doran et al., 6 Aug 2024).

The Gross–Siebert program leverages Tyurin degenerations to construct walls (real codimension one loci in the base of torus fibrations), arising from families of holomorphic disks whose boundaries lie on Lagrangian tori. These walls are loci where the count of Maslov index 2 disks changes, controlling scattering diagrams and wall-crossing phenomena critical for mirror symmetry (Nishinou, 2011).

4. Lattice Theory, Mirror Symmetry, and Fibration Correspondence

Tyurin degenerations are crucial for lattice polarized mirror symmetry, especially for K3 surfaces. For a degeneration with central fiber $X_0 = V_1 \cup_C V_2$ , one studies the numerical Grothendieck groups $K_0^{num}(D(V_i))$ , using spherical homomorphisms and pseudolattice formalisms to glue the Néron–Severi lattices along the elliptic pseudolattice $E$ defined by generators $a, b$ with anti-symmetric pairing. The global lattice

$NS(M) \simeq H \oplus E_8 \oplus E_8$

emerges as the mirror lattice, compatible with Dolgachev–Nikulin mirror symmetry (Giovenzana et al., 20 May 2024, Doran et al., 2023).

On the mirror side, one considers elliptically fibered K3 surfaces whose base splits into two discs reflecting the decomposition of the Tyurin central fiber; the matching of relative homology and polarizing lattices on both sides establishes a precise correspondence. In higher dimensions, the Doran–Harder–Thompson conjecture asserts that the mirror of a Calabi–Yau admitting a Tyurin degeneration is constructed by gluing the Landau–Ginzburg mirrors of its quasi-Fano components along their shared divisor (Kanazawa, 2018, Doran et al., 2016).

5. Derived Geometry, Moduli Spaces, and Categorified DT Invariants

Recent advances extend Tyurin degeneration to the field of derived algebraic geometry and categorification. The moduli space of perfect complexes (or rigidified objects) on a degenerating Calabi–Yau threefold

$X \rightsquigarrow X_1 \cup_S X_2$

degenerates into the derived fiber product

$M(X_1) \times_{M(S)} M(X_2)$

where $M(X_i)$ are moduli spaces over the quasi-Fano components and $M(S)$ over the common anti-canonical divisor (Kryczka et al., 23 Oct 2025). Each Fano moduli space maps as a derived Lagrangian into $M(S)$ ; their derived intersection computes categorified DT invariants via matrix factorization categories and periodic cyclic homology. The existence of a flat Gauss–Manin connection on the periodic cyclic homology ensures deformation invariance of the categorified invariants across the degeneration parameter, and the derived intersection cohomology glues invariants from the Fano moduli spaces to those on the smooth Calabi–Yau.

6. Significance for Moduli, Compactifications, and Geometry

Tyurin degenerations classify and parametrize boundary components in the compactification of moduli spaces—for instance, the Baily–Borel compactification of polarized K3 surfaces (Jones, 6 Feb 2025). Explicit families of Tyurin degenerations account for each boundary stratum and yield stable models via flop and contraction analysis based on lifted polarizations. Projectivity is shown not to restrict the spectrum of central fiber components: every possible weak del Pezzo degree can be realized in projective models for M-polarized K3 Tyurin degenerations (Doran et al., 2023).

The technical interplay between degeneration geometry, log deformation theory, modular forms, lattice theory, and derived categories situates Tyurin degeneration as a unifying framework for modern algebraic geometry, enumerative theory, mirror symmetry, and category theory.

Table: Key Features of Tyurin Degenerations (Selected Contexts)

Context	Central Fiber Structure	Notable Application
K3 surface (Giovenzana et al., 20 May 2024)	Two rational surfaces glued along an elliptic curve	Mirror symmetry via lattice polarization
Calabi–Yau threefold (Nishinou, 2011)	Union of toric Fano components, e.g., projective spaces glued along toric strata	Combinatorial curve counting; derivation of 2875 lines on quintic
Complete intersections (Kryczka et al., 23 Oct 2025)	Union of quasi-Fano threefolds meeting along anti-canonical divisor	Derived Lagrangian intersections, categorification of DT invariants

7. Concluding Remarks

Tyurin degeneration provides a concrete geometric framework for the paper of degenerations in complex algebraic geometry, with ramifications for curve counting, deformation theory, moduli compactification, wall structures in mirror symmetry, lattice theory, categorical invariants, and modularity of enumerative invariants. The process reduces intricate deformation problems to explicit local computations on reducible singular fibers, bridges classical and modern mirror symmetry (including Landau–Ginzburg and derived categorical perspectives), and offers robust tools for both theoretical and computational advancements in algebraic geometry and its applications.