Ordinary Gushel–Mukai Threefolds

• Ordinary Gushel–Mukai threefolds are smooth prime Fano threefolds defined as quadric sections of a codimension-2 linear section of Gr(2,5), characterized by Picard rank 1, degree 10, and genus 6.
• They serve as a rich testing ground for categorical Torelli theorems, Bridgeland stability conditions, and period map studies, linking classical algebraic geometry with modern derived category techniques.
• Their structure, involving Lagrangian EPW data and Kuznetsov components, governs deep birational properties and moduli phenomena, providing insights into noncommutative and hyperkähler geometry.

An ordinary Gushel–Mukai threefold is a smooth prime Fano threefold of genus $6$, equivalently a smooth Fano threefold of Picard rank $1$, index $1$, and degree $10$, realized as a quadric section of a codimension-$2$ linear section of $\operatorname{Gr}(2,5)$ in its Plücker embedding. In the modern formulation, it sits simultaneously in the geometry of linear sections of Grassmannians, the EPW–Lagrangian correspondence, and the study of nontrivial semiorthogonal components of derived categories. Ordinary GM threefolds provide the three-dimensional member of the Gushel–Mukai hierarchy and are a central testing ground for questions on period maps, categorical Torelli theorems, Bridgeland stability, and birational transformations (Debarre et al., 2015, Debarre, 2020).

1. Geometric definition and basic invariants

Let $V_5$ be a $5$-dimensional vector space and let $$\operatorname{Gr}(2,V_5)\subset \mathbf P(\wedge^2 V_5)\cong \mathbf P^9$$ be the Plücker-embedded Grassmannian. An ordinary Gushel–Mukai threefold is a smooth variety of the form

$$X=\operatorname{Gr}(2,5)\cap \mathbf P^7\cap Q \subset \mathbf P^9,$$

where $1$0 is a codimension-$1$1 linear subspace and $1$2 is a quadric hypersurface. Equivalently, in the cone model, a GM variety is

$1$3

and the variety is ordinary precisely when the linear space $1$4 does not contain the cone vertex; in dimension $1$5, this recovers the Grassmannian model above (Debarre et al., 2015, Kuznetsov et al., 2016).

For a GM threefold $1$6, the polarization $1$7 satisfies

$1$8

Hence $1$9 is a prime Fano threefold of degree $1$0 and genus $1$1. The ordinary/special dichotomy is intrinsic: ordinary GM data are characterized by the injectivity of the map defining the Gushel structure, whereas special GM threefolds are double covers of codimension-$1$2 linear sections of $1$3 branched along a K3 surface or, in the cone language, correspond to linear spaces through the cone vertex (Debarre et al., 2015, Lin et al., 9 Jul 2025).

The Gushel morphism

$1$4

is a closed embedding in the ordinary case. Pulling back the tautological rank-$1$5 subbundle on $1$6 gives the Gushel bundle. In one common notation, if $1$7 is the tautological subbundle, then

$1$8

and this exceptional rank-$1$9 bundle plays a distinguished role in the derived category and in refined categorical reconstruction (Jacovskis et al., 2021).

2. Lagrangian data, EPW geometry, and moduli

Ordinary GM threefolds admit a uniform description in terms of Lagrangian data $10$0, where $10$1 is $10$2-dimensional, $10$3 is a hyperplane, and $10$4 is a Lagrangian subspace. For ordinary GM threefolds one has

$10$5

and, in the smooth threefold case, the relevant Lagrangian $10$6 contains no decomposable vectors (Debarre et al., 2016, Debarre et al., 2015).

To a smooth GM threefold one associates the $10$7-dimensional space of quadrics $10$8 containing $10$9, together with the Plücker hyperplane $2$0. The associated EPW sextic is

$2$1

and, for smooth ordinary GM varieties of dimension $2$2, Debarre’s survey states that $2$3 is identified with the discriminant locus of singular quadrics through $2$4 after removing the Plücker hyperplane with the appropriate multiplicity. For odd-dimensional GM varieties, the surface $2$5 and its canonical double cover $2$6 are especially important: for $2$7,

$2$8

as principally polarized abelian varieties, where $2$9 is the intermediate Jacobian of $\operatorname{Gr}(2,5)$0 (Debarre, 2020).

The moduli space of smooth GM threefolds is irreducible of dimension $\operatorname{Gr}(2,5)$1. In the survey formulation,

$\operatorname{Gr}(2,5)$2

and the map to EPW moduli has fibers described by EPW strata: $\operatorname{Gr}(2,5)$3 The ordinary locus is encoded by the appropriate EPW stratum, while special threefolds occur on the adjacent stratum. The same Lagrangian $\operatorname{Gr}(2,5)$4 can therefore underlie different GM threefolds, giving rise to the notions of period partners and dual varieties (Debarre, 2020, Debarre et al., 2015).

This framework has direct birational consequences. Period partners of the same dimension share the same EPW sextic, and in dimension $\operatorname{Gr}(2,5)$5 smooth period partners are birationally isomorphic. Smooth dual GM threefolds are also birationally isomorphic. A plausible implication is that the Lagrangian $\operatorname{Gr}(2,5)$6 governs the birational geometry of ordinary GM threefolds more rigidly than the ambient Grassmannian presentation alone (Debarre et al., 2015).

3. Derived categories, Kuznetsov components, and Bridgeland stability

For a GM threefold $\operatorname{Gr}(2,5)$7, Kuznetsov–Perry isolate a nontrivial admissible component of the derived category. In the notation of "Derived categories of Gushel-Mukai varieties" one has

$\operatorname{Gr}(2,5)$8

where $\operatorname{Gr}(2,5)$9 is the pullback of the tautological rank-$V_5$0 bundle on $V_5$1, and $V_5$2 is the GM category. For threefolds, this category has Serre functor

$V_5$3

with $V_5$4 a nontrivial involution, so the odd-dimensional GM category is described there as Enriques-type. Its Hochschild invariants are

$V_5$5

and

$V_5$6

with Euler form $V_5$7 in a suitable basis (Kuznetsov et al., 2016).

Later papers use closely related notations $V_5$8, $V_5$9, and $5$0. For a GM threefold they write

$5$1

and compute explicit numerical lattices of rank $5$2. One basis is

$5$3

with Euler form $5$4; another basis is

$5$5

with Euler matrix

$5$6

The supplied literature therefore treats the same surface-like nontrivial component through several equivalent models adapted to stability, mutations, or Torelli arguments (Pertusi et al., 2021, Jacovskis et al., 2021).

Bridgeland stability conditions on the Kuznetsov component of a GM threefold are constructed by restricting double-tilt stability on $5$7. The stability-condition literature proves that the resulting conditions are preserved by the Serre functor up to the $5$8-action, and that there is a unique $5$9-orbit of Serre-invariant stability conditions on the GM-threefold Kuznetsov component (Pertusi et al., 2021). For general GM threefolds, moduli spaces

$$\operatorname{Gr}(2,V_5)\subset \mathbf P(\wedge^2 V_5)\cong \mathbf P^9$$0

of $$\operatorname{Gr}(2,V_5)\subset \mathbf P(\wedge^2 V_5)\cong \mathbf P^9$$1-semistable objects of class $$\operatorname{Gr}(2,V_5)\subset \mathbf P(\wedge^2 V_5)\cong \mathbf P^9$$2 are projective schemes, confirming the projectivity expected in earlier work (Feyzbakhsh et al., 2024).

4. Period maps, infinitesimal Torelli, and categorical reconstruction

The Hodge-theoretic period map for ordinary GM threefolds takes values in $$\operatorname{Gr}(2,V_5)\subset \mathbf P(\wedge^2 V_5)\cong \mathbf P^9$$3, the moduli of principally polarized abelian $$\operatorname{Gr}(2,V_5)\subset \mathbf P(\wedge^2 V_5)\cong \mathbf P^9$$4-folds, via the intermediate Jacobian. The period map is highly non-injective: the 2025 infinitesimal Torelli study recalls the result of Debarre–Iliev–Manivel that for an ordinary GM threefold the differential of the period map has a $$\operatorname{Gr}(2,V_5)\subset \mathbf P(\wedge^2 V_5)\cong \mathbf P^9$$5-dimensional kernel. The same paper identifies this kernel categorically: $$\operatorname{Gr}(2,V_5)\subset \mathbf P(\wedge^2 V_5)\cong \mathbf P^9$$6 and

$$\operatorname{Gr}(2,V_5)\subset \mathbf P(\wedge^2 V_5)\cong \mathbf P^9$$7

is identified with the exceptional locus of the birational contraction from the Fano surface of conics to a Bridgeland moduli space in the Kuznetsov component (Lin et al., 9 Jul 2025).

Categorical Torelli results sharpen this picture. For a general ordinary GM threefold $$\operatorname{Gr}(2,V_5)\subset \mathbf P(\wedge^2 V_5)\cong \mathbf P^9$$8, the Kuznetsov component together with the projected tautological bundle determines the variety: $$\operatorname{Gr}(2,V_5)\subset \mathbf P(\wedge^2 V_5)\cong \mathbf P^9$$9 More precisely, if $$X=\operatorname{Gr}(2,5)\cap \mathbf P^7\cap Q \subset \mathbf P^9,$$0 sends $$X=\operatorname{Gr}(2,5)\cap \mathbf P^7\cap Q \subset \mathbf P^9,$$1 to $$X=\operatorname{Gr}(2,5)\cap \mathbf P^7\cap Q \subset \mathbf P^9,$$2, then $$X=\operatorname{Gr}(2,5)\cap \mathbf P^7\cap Q \subset \mathbf P^9,$$3. Without the extra object, the Kuznetsov component determines the birational class of a general ordinary GM threefold. For general special GM threefolds, $$X=\operatorname{Gr}(2,5)\cap \mathbf P^7\cap Q \subset \mathbf P^9,$$4 alone determines the isomorphism class (Jacovskis et al., 2021).

The same paper reformulates the Debarre–Iliev–Manivel conjecture on fibers of the classical period map by introducing a categorical period map whose fiber over $$X=\operatorname{Gr}(2,5)\cap \mathbf P^7\cap Q \subset \mathbf P^9,$$5 is the disjoint union

$$X=\operatorname{Gr}(2,5)\cap \mathbf P^7\cap Q \subset \mathbf P^9,$$6

where $$X=\operatorname{Gr}(2,5)\cap \mathbf P^7\cap Q \subset \mathbf P^9,$$7 is the minimal model of the Fano surface of conics and $$X=\operatorname{Gr}(2,5)\cap \mathbf P^7\cap Q \subset \mathbf P^9,$$8 is a rank-$$X=\operatorname{Gr}(2,5)\cap \mathbf P^7\cap Q \subset \mathbf P^9,$$9 sheaf moduli space. In the general ordinary case, the conjectural fiber of the classical period map is therefore recast as the fiber of a categorical map governed by the Kuznetsov component (Jacovskis et al., 2021).

5. Lines, conics, twisted cubics, and quadrics

The low-degree curve geometry of ordinary GM threefolds is unusually rich. For a general GM threefold, the Hilbert scheme of lines $1$00 is a smooth connected curve of genus $1$01, and the natural map to the EPW surface is the normalization of a $1$02-nodal curve inside $1$03 (Debarre, 2020).

Conics form the most classical surface-type parameter space. Debarre’s survey reports Logachev’s result that the Hilbert scheme of conics on a GM threefold is the blow-up of $1$04 at one point (Debarre, 2020). A later analysis of quadrics on GM varieties gives, for a smooth ordinary GM threefold $1$05, a more explicit description: $1$06 the blowup of the double dual EPW surface at one point lying over the Plücker point. In the same framework, $1$07 is a connected Cohen–Macaulay surface of pure dimension $1$08, and GM threefolds contain no planes: $1$09 The construction proceeds through a family of conics $1$10 over

$1$11

whose corank stratification matches the blowup of the dual EPW stratification (Debarre et al., 2024).

Twisted cubics on general GM threefolds provide a further bridge to derived geometry. For a GM threefold $1$12, the Hilbert scheme $1$13 parametrizes Cohen–Macaulay twisted cubics. For a general $1$14, it is a smooth irreducible threefold. Projection to the Kuznetsov component gives a morphism

$1$15

which is an open immersion away from a locus $1$16, and $1$17 is a $1$18-bundle over the Hilbert scheme of lines. The image $1$19 is identified with the Hilbert scheme of lines, so the Bridgeland moduli space $1$20 is a threefold birational to the Hilbert scheme of twisted cubics, with the $1$21-families of residue cubics contracted along the line locus (Feyzbakhsh et al., 22 Jan 2025).

These curve spaces interact directly with hyperkähler geometry in one higher dimension. If $1$22 is a hyperplane section of a general GM fourfold $1$23, the Bridgeland moduli spaces on $1$24 map to moduli on $1$25, and for the class $1$26 the images are Lagrangian subvarieties in the hyperkähler sixfold associated with twisted cubics on $1$27. A family of such GM-threefold hyperplane sections gives a Lagrangian covering family of the double EPW cube (Feyzbakhsh et al., 22 Jan 2025, Feyzbakhsh et al., 2024).

6. Birational transformations, explicit examples, singular models, and stability

Ordinary GM threefolds are linked by birational operations known as conic transforms and line transforms. The classification-and-birationalities paper proves that smooth period partners and smooth dual GM varieties of the same dimension are birationally isomorphic. In dimension $1$28, this yields infinitely many non-isomorphic prime Fano threefolds birationally equivalent to a given ordinary GM threefold (Debarre et al., 2015).

Rationality questions remain a major theme. Debarre’s survey states that a general GM threefold is not rational and records the expectation that all smooth GM threefolds should be irrational, while noting that a full proof for all smooth members was not available there (Debarre, 2020). An explicit ordinary GM threefold with a faithful $1$29-action was later constructed and proved irrational by analyzing its intermediate Jacobian; the same paper exhibits a complete $1$30-dimensional family of irrational smooth ordinary GM threefolds associated with the same Lagrangian $1$31 (Debarre et al., 2021).

The singular, nodal case has its own birational-categorical structure. A $1$32-nodal ordinary GM threefold

$1$33

with a unique node $1$34 admits a blowup $1$35 and a relative Atiyah flop to a conic bundle

$1$36

with smooth sextic discriminant. The categorical resolution of the Kuznetsov component is then identified with the Clifford category

$1$37

and this categorical resolution determines the birational class of a general $1$38-nodal ordinary GM threefold (Grzelakowski et al., 15 Feb 2026).

K-stability adds another moduli-theoretic layer. The K-stability paper recalls that all smooth GM threefolds are K-stable and studies ordinary GM threefolds as divisors

$1$39

on the quintic del Pezzo fourfold $1$40. For a general ordinary GM threefold $1$41, the K-semistable domain of the pair is

$1$42

This interval controls the K-stability of special GM fourfolds realized as double covers of $1$43 branched along $1$44, and shows that ordinary GM threefolds occupy a natural boundary position in K-moduli wall-crossing (Liu et al., 2024).

Taken together, these results place ordinary Gushel–Mukai threefolds at a junction where Grassmannian linear-section geometry, EPW degeneracy theory, noncommutative surface-like categories, Bridgeland moduli, and birational transformations all interact through a common rank-$1$45 categorical core.