Special Gushel–Mukai Threefolds
- Special Gushel–Mukai threefolds are smooth Fano threefolds of genus 6 with Picard rank 1 and degree 10, constructed as double covers of linear sections of Gr(2,5) with a K3 branch surface.
- Their geometry is governed by a covering involution that impacts Hodge theory and deformation properties, leading to a distinct infinitesimal Torelli behavior compared to ordinary cases.
- The derived categories of these threefolds feature an Enriques-type Kuznetsov component, underpinning a categorical Torelli theorem and linking them to associated K3 surfaces and EPW degeneracy loci.
Special Gushel–Mukai threefolds are the special members of the genus-$6$ prime Fano family: smooth Fano threefolds with Picard rank $1$, index $1$, and degree $10$, realized not as ordinary quadric sections of linear sections of , but as double covers of a codimension-$3$ linear section of branched along a smooth quadric section, equivalently along a degree-$10$ surface (Lin et al., 9 Jul 2025, Debarre et al., 2015). Their geometry is governed by the covering involution, their Hodge theory differs sharply from the ordinary case, and their derived categories exhibit an Enriques-type structure whose equivariant and covering constructions recover the branch surface.
1. Definition and basic geometry
A Gushel–Mukai variety is a smooth intersection
$1$0
where $1$1 is the projective cone over the Plücker Grassmannian and $1$2 is a quadric hypersurface. The ordinary/special dichotomy is determined by the cone vertex: a GM variety is ordinary if the relevant linear section avoids the vertex, and special if it contains the vertex (Debarre, 2020, Liu et al., 2024).
In dimension $1$3, a GM threefold is a smooth Fano threefold with
$1$4
hence a prime Fano threefold of genus $1$5 (Liu et al., 2024, Debarre, 2020). The ordinary model is
$1$6
By contrast, a special Gushel–Mukai threefold is a double cover
$1$7
of the codimension-$1$8 linear section
$1$9
branched along a smooth quadric section
$1$0
If $1$1 denotes the hyperplane class on $1$2, then
$1$3
and $1$4 is a smooth $1$5 surface of degree $1$6 (Lin et al., 9 Jul 2025).
The same threefold $1$7 is described in the $1$8-stability literature as the unique quintic del Pezzo threefold $1$9, characterized by
$10$0
With $10$1 the branch divisor, one has
$10$2
so $10$3, in agreement with the index-$10$4 Fano structure (Liu et al., 2024). In this form, a special GM threefold is simultaneously a branched double cover of a degree-$10$5 del Pezzo threefold and a member of the standard GM family.
2. Lagrangian data, EPW geometry, and the special locus
The modern classification of GM varieties uses Lagrangian data. For lci GM varieties, one passes from GM data to an extended Lagrangian data set
$10$6
where $10$7 is Lagrangian and $10$8 encodes the type: changing $10$9 switches between ordinary and special (Debarre et al., 2015). The dimension formula is
0
Hence a special GM threefold satisfies
1
whereas an ordinary GM threefold satisfies
2
(Debarre et al., 2015, Liu et al., 15 Dec 2025).
Special and ordinary GM varieties are related by the opposite construction. If 3 is a special GM threefold, then its opposite variety 4 is the branch divisor of the double cover, hence an ordinary GM surface; in the threefold case this is a polarized 5 surface of degree 6 (Bayer et al., 2022). This relation is central: the special threefold and the ordinary surface share the same Lagrangian/EPW data, but differ by type.
The EPW description makes the moduli-theoretic position of the special locus precise. For GM threefolds, the fiber of the natural map from GM moduli to EPW moduli has the form
7
where the piece 8 parametrizes ordinary GM threefolds and the piece 9 parametrizes special GM threefolds (Debarre, 2020). Thus special GM threefolds form the deeper degeneracy stratum in the EPW picture. At the same time, a special GM threefold has the same corrected discriminant and hence the same EPW sextic as its associated ordinary partner (Debarre et al., 2015). This indicates that the special character is not detected by the sextic $3$0 alone; it depends on the type datum and the specific position of $3$1.
3. Hodge theory and infinitesimal Torelli
For special GM threefolds, the covering involution $3$2 fundamentally alters the infinitesimal Torelli problem. The infinitesimal period map is
$3$3
and the double-cover structure makes it $3$4-equivariant. Using Konno’s decomposition for double covers, one obtains
$3$5
together with corresponding splittings for the Hodge spaces. In the special GM case, the anti-invariant part of the infinitesimal period map is zero, and
$3$6
Moreover,
$3$7
so the invariant part has dimension $3$8 (Lin et al., 9 Jul 2025).
The main infinitesimal Torelli theorem is that the invariant part
$3$9
is injective. Consequently,
0
(Lin et al., 9 Jul 2025). This is the precise infinitesimal Torelli statement for special GM threefolds: 1 is not injective, but its invariant part is injective, and the entire failure of injectivity comes from the anti-invariant directions produced by the covering involution.
The geometric interpretation is equally specific. The invariant subspace
2
parametrizes first-order deformations preserving the structure of 3 as a special GM threefold, namely deformations of the branch 4 surface 5 inside the fixed rigid threefold 6. The 7-dimensional kernel consists of normal directions leaving the special locus toward possibly ordinary GM threefolds (Lin et al., 9 Jul 2025). This is why the paper treats special GM threefolds as hyperelliptic-type members of the genus-8 family: the involution contributes precisely the extra invisible directions.
The proof reduces the problem to a twisted infinitesimal Torelli statement on the branch 9 surface. One has an injection
$10$0
with image a $10$1-dimensional subspace $10$2, while
$10$3
The missing $10$4-dimensional direction corresponds to the polarization inherited from $10$5, and the unrestricted map on all of $10$6 is not injective (Lin et al., 9 Jul 2025). This sharply distinguishes the special GM Torelli statement from the usual $10$7 infinitesimal Torelli theorem.
4. Derived categories, Enriques structure, and categorical Torelli
For a GM threefold $10$8, the derived category admits a semiorthogonal decomposition
$10$9
and equivalently
0
where 1 is the pullback of the tautological rank-2 subbundle on 3 and 4 (Kuznetsov et al., 2016, Jacovskis et al., 2021). The nontrivial component is not geometric in the sense of being 5 for a variety 6: for odd-dimensional GM varieties, and hence for GM threefolds, the GM category is an Enriques category rather than a K3 category (Kuznetsov et al., 2016).
More precisely, for odd-dimensional GM varieties one has
7
with 8 a nontrivial involution. If 9 is special, this involution is induced by the covering involution of the double cover. In the threefold notation of the categorical Torelli paper,
0
for a special GM threefold 1 with covering involution 2 (Kuznetsov et al., 2016, Jacovskis et al., 2021). This is the categorical signature of specialness: the Enriques involution is geometric.
The same phenomenon appears in covering constructions. For a special GM threefold, the equivariant Kuznetsov component recovers the branch surface: 3 where 4 is the branch 5 surface (Jacovskis et al., 2021). Equivalently, the CY6 cover of the Enriques category 7 is the derived category of the opposite GM surface 8, a degree-9 $1$00 surface (Bayer et al., 2022). This identifies special GM threefolds as noncommutative Enriques objects with a genuine $1$01 cover.
This structure yields a categorical Torelli theorem that is stronger in the special case than in the ordinary one. For general special GM threefolds $1$02 and $1$03,
$1$04
(Jacovskis et al., 2021). The proof uses the identity $1$05: any equivalence of Kuznetsov components must commute with Serre functors, hence with the involutions, so it descends to an equivalence of the equivariant categories and therefore of the derived categories of the branch $1$06 surfaces. Since the branch surfaces are degree-$1$07, Picard-rank-$1$08 $1$09 surfaces, derived equivalence implies isomorphism, and the rigidity of the ambient del Pezzo threefold recovers the double cover (Jacovskis et al., 2021).
By contrast, the ordinary case is less rigid. For general ordinary GM threefolds, the Kuznetsov component alone generally determines only the birational class, and a refined Torelli statement requires an additional distinguished object (Jacovskis et al., 2021). The broader categorical classification of GM threefolds states that equivalent Kuznetsov components occur exactly for period partners or duals, and in particular imply birationality; special points are included in that classification (Bayer et al., 2022). The special case is distinguished because the involution upgrades birational categorical Torelli to isomorphism-level categorical Torelli.
5. Bridgeland stability, Kuznetsov moduli, and EPW surfaces
Bridgeland stability conditions exist on the Kuznetsov component of every GM variety, hence in particular on the Kuznetsov component of every special GM threefold (Perry et al., 2019). For GM threefolds, the stability conditions constructed by Bayer, Lahoz, Macrì, and Stellari are preserved by the Serre functor up to the natural $1$10-action, and in fact
$1$11
for stability conditions in the relevant orbit (Pertusi et al., 2021). The same paper proves that all Serre-invariant stability conditions on the Kuznetsov component of a GM threefold lie in a single $1$12-orbit (Pertusi et al., 2021). For a special GM threefold, this Serre invariance is the categorical form of invariance under the covering involution.
When $1$13 is the special GM threefold associated with a strongly smooth ordinary GM surface $1$14, the equivariant relation between $1$15 and $1$16 becomes an explicit moduli machine. The numerical Grothendieck group of $1$17 has rank $1$18, with basis $1$19 satisfying
$1$20
and Euler form $1$21, where $1$22. Under the inflation map, the classes $1$23 and $1$24 correspond to the Mukai vectors
$1$25
on the branch surface $1$26 (Liu et al., 15 Dec 2025). The induced stability condition $1$27 on $1$28 then yields two distinguished moduli spaces: $1$29 Thus the double dual EPW surface and the double EPW surface associated with a special GM threefold are realized as moduli spaces of semistable objects on its Kuznetsov component (Liu et al., 15 Dec 2025). These identifications refine the Bayer–Perry statement on equivalent Kuznetsov components: the action on $1$30 distinguishes period-partner behavior from period-dual behavior (Liu et al., 15 Dec 2025).
The Hilbert schemes of curves on special GM threefolds also interact nontrivially with Bridgeland moduli. For a special smooth GM threefold $1$31, the Hilbert scheme
$1$32
of twisted cubics is a smooth irreducible projective threefold if $1$33 is general among special GM threefolds, and singular if $1$34 is not general. In the general case there exists an irreducible component $1$35 of a moduli space of Bridgeland stable objects in $1$36 such that
$1$37
is a divisorial contraction (Zhang, 2020). The special geometry of the branch divisor controls the singularities in the nongeneral case.
A different projectivity result applies to a general GM threefold $1$38. For coprime integers $1$39 and any Serre-invariant stability condition $1$40 on $1$41, the moduli space
$1$42
is a projective scheme (Feyzbakhsh et al., 2024). The paper explicitly presents this theorem for a general GM threefold and notes that this class can be either ordinary or special. The proof uses a finite morphism from the threefold moduli space to a projective moduli space on a general GM fourfold containing $1$43 as a hyperplane section (Feyzbakhsh et al., 2024). This places special GM threefolds inside a broader hyperplane-section comparison framework between Enriques-type and K3-type Kuznetsov components.
6. Birationality, $1$44-stability, and terminological clarifications
In the EPW/Lagrangian theory, special GM threefolds participate in the same birational web as ordinary ones. Smooth period partners of dimension $1$45 are birationally isomorphic, and smooth dual GM threefolds are also birationally isomorphic (Debarre et al., 2015). Combined with the later categorical classification of GM threefolds, this yields a stable picture: equivalence of Kuznetsov components implies period-partner or dual behavior and hence birationality, while the special case is sharpened by the categorical Torelli theorem to actual isomorphism for general special threefolds (Bayer et al., 2022, Jacovskis et al., 2021).
From the viewpoint of $1$46-stability, the threefold case is already settled in the literature cited by recent work: all smooth GM threefolds are $1$47-stable. For a special GM threefold $1$48, the log-pair description
$1$49
identifies special GM threefolds with the log Fano pair $1$50, and
$1$51
(Liu et al., 2024). The 2024 $1$52-stability paper does not prove a new threefold theorem; rather, it records that the stronger statement “all smooth GM threefolds are $1$53-stable” was already known and uses the double-cover/log-pair framework as the three-dimensional prototype for higher-dimensional special GM manifolds (Liu et al., 2024).
A recurrent terminological ambiguity deserves explicit separation. In the standard Mukai–Debarre–Kuznetsov sense, a special GM threefold is the double-cover type described above. By contrast, a 1-nodal ordinary GM threefold is a singular ordinary GM threefold, not a special GM variety in the standard sense. Recent work on nodal GM varieties studies this singular locus as a distinguished degeneration of ordinary GM threefolds and develops its own conic-bundle and Clifford-algebra description, but it explicitly notes that these are not special GM varieties in the usual classification terminology (Grzelakowski et al., 15 Feb 2026). This distinction matters because both settings involve extra involutions and special birational structures, but they belong to different strata of the GM landscape.
Special Gushel–Mukai threefolds therefore occupy a sharply defined position at the intersection of Fano geometry, $1$54 surfaces, EPW degeneracy loci, and noncommutative surface theory. Their defining double-cover involution governs their deformation theory, explains the precise form of infinitesimal Torelli, endows their Kuznetsov components with geometric Enriques structure, and makes possible a categorical Torelli theorem that is stronger than in the ordinary case.
