Mori Dream K3 Surface: Geometry and Cox Rings

• Mori dream K3 surfaces are K3 surfaces with finitely generated Cox rings, equating to finite automorphism groups and rational polyhedral effective cones.
• They connect lattice theory, cone geometry, and projective models through detailed classifications of the Néron–Severi lattice and explicit geometric constructions.
• Explicit models include quartics, double covers, and hypersurfaces in ambient Mori dream spaces, providing actionable insights into their algebraic and geometric structure.

A Mori dream K3 surface is a K3 surface $X$ for which the Cox ring

$$R(X)=\bigoplus_{D\in K} H^0\bigl(X,\mathcal O_X(D)\bigr)$$

is a finitely generated $\mathbb C$-algebra, equivalently a K3 surface that is a Mori dream space in the sense of Hu–Keel and ADHL. For K3 surfaces this notion admits unusually strong reformulations: finite generation of the Cox ring, rational polyhedrality of the effective cone, and finiteness of the automorphism group are equivalent. The subject therefore lies at the intersection of lattice theory, cone geometry, projective models, and explicit multigraded algebra. In the smooth case the decisive invariant is the Néron–Severi lattice; in the singular case rational double points and their resolutions enter through the Picard lattice and the behavior of negative curves (Artebani et al., 2020, Laface et al., 2024).

1. Definitions, cones, and equivalent criteria

For a normal projective variety $X$ over $\mathbb C$ with finitely generated, free divisor class group $\mathrm{Cl}(X)$, the Cox ring is the total coordinate ring

$$R(X)=\bigoplus_{D\in K} H^0\bigl(X,\mathcal O_X(D)\bigr),$$

graded by $K$, equivalently by $\mathrm{Cl}(X)$. A projective variety is called a Mori dream space if $R(X)$ is finitely generated. In the smooth projective situation recalled in the Picard-rank-four study, this is equivalent to the effective cone $$R(X)=\bigoplus_{D\in K} H^0\bigl(X,\mathcal O_X(D)\bigr)$$0 being rational polyhedral (Artebani et al., 2020).

For a K3 surface, $$R(X)=\bigoplus_{D\in K} H^0\bigl(X,\mathcal O_X(D)\bigr)$$1, and the defining condition becomes especially rigid. The cited theorem specialized to K3 surfaces states that for an algebraic K3 surface the following are equivalent: $$R(X)=\bigoplus_{D\in K} H^0\bigl(X,\mathcal O_X(D)\bigr)$$2 is a Mori dream surface, the effective cone is polyhedral, and $$R(X)=\bigoplus_{D\in K} H^0\bigl(X,\mathcal O_X(D)\bigr)$$3 is finite. Thus

$$R(X)=\bigoplus_{D\in K} H^0\bigl(X,\mathcal O_X(D)\bigr)$$4

When $$R(X)=\bigoplus_{D\in K} H^0\bigl(X,\mathcal O_X(D)\bigr)$$5 and the automorphism group is finite, the surface has only finitely many $$R(X)=\bigoplus_{D\in K} H^0\bigl(X,\mathcal O_X(D)\bigr)$$6-curves, and the classes of $$R(X)=\bigoplus_{D\in K} H^0\bigl(X,\mathcal O_X(D)\bigr)$$7-curves generate the effective cone (Artebani et al., 2020).

This equivalence connects birational geometry to lattice theory through the global Torelli theorem. If $$R(X)=\bigoplus_{D\in K} H^0\bigl(X,\mathcal O_X(D)\bigr)$$8, $$R(X)=\bigoplus_{D\in K} H^0\bigl(X,\mathcal O_X(D)\bigr)$$9 is its isometry group, and $\mathbb C$0 is generated by reflections in classes of square $\mathbb C$1, then $\mathbb C$2 is finite if and only if $\mathbb C$3 is finite. A hyperbolic lattice with this property is called 2-reflective. For Mori dream K3 surfaces, the cone geometry is therefore encoded by a hyperbolic lattice with finitely many relevant root walls (Artebani et al., 2020).

In Picard number $\mathbb C$4, the criterion becomes more elementary. For smooth K3 surfaces, the cited rank-two characterization states that $\mathbb C$5 is Mori dream if and only if $\mathbb C$6 contains a class of self-intersection $\mathbb C$7 or $\mathbb C$8, equivalently there is an elliptic fibration or a $\mathbb C$9-curve. For possibly singular projective K3 surfaces with $X$0, a later refinement gives an equivalent intrinsic condition: $X$1 is Mori dream if and only if there exist two effective divisors $X$2 such that $X$3 and $X$4 (Garbagnati, 2015, Laface et al., 2024).

2. Lattice-theoretic structure and classification phenomena

The Néron–Severi lattice of a complex K3 surface is an even lattice of signature $X$5. For Mori dream K3 surfaces, this lattice controls both automorphisms and the effective cone. The Picard-rank-four classification makes this explicit: if $X$6 is an algebraic complex K3 surface with $X$7 and finite automorphism group, then $X$8 is isometric to one of $X$9 lattices $\mathbb C$0, listed explicitly as

$\mathbb C$1

together with the explicitly displayed matrices $\mathbb C$2 and $\mathbb C$3. Hence a K3 surface with $\mathbb C$4 is a Mori dream K3 surface if and only if its Néron–Severi lattice is one of these $\mathbb C$5 Vinberg lattices (Artebani et al., 2020).

For each such lattice $\mathbb C$6, the family $\mathbb C$7 consists of K3 surfaces with $\mathbb C$8, and the corresponding moduli space $\mathbb C$9 has dimension $\mathrm{Cl}(X)$0. The same work computes, for every $\mathrm{Cl}(X)$1, the set $\mathrm{Cl}(X)$2 of extremal rays of the effective cone, identifies these rays with classes of $\mathrm{Cl}(X)$3-curves, and gives explicit Hilbert bases for both the effective and nef cones. This Hilbert-basis data is then used to bound and in many cases determine the degrees of minimal generators of the Cox ring (Artebani et al., 2020).

The singular theory introduces an allied but distinct lattice notion. A sublattice $\mathrm{Cl}(X)$4 is called a Mori dream lattice if there exists a smooth K3 surface with Picard lattice isometric to $\mathrm{Cl}(X)$5 that is a Mori dream space. The main structural statement in the singular setting is that if the Picard lattice of a singular K3 surface is Mori dream, then the surface itself is Mori dream. This transfers the smooth lattice classification into a criterion for singular K3 surfaces with rational double points (Laface et al., 2024).

A plausible implication is that the smooth and singular theories are organized by the same discrete lattice data, but with different geometric realizations: smooth models are controlled directly by $\mathrm{Cl}(X)$6, while singular models are controlled by the pullback lattice inside the Picard lattice of the minimal resolution.

3. Projective models and explicit Cox rings

For Picard number $\mathrm{Cl}(X)$7, the lattice classification is accompanied by a detailed geometric classification of projective models. Each family $\mathrm{Cl}(X)$8 admits an explicit realization, such as complete intersections of quadrics, quartic surfaces with prescribed reducible hyperplane sections, or double covers of $\mathrm{Cl}(X)$9 or Hirzebruch surfaces with controlled branch data. Examples include:

• $$R(X)=\bigoplus_{D\in K} H^0\bigl(X,\mathcal O_X(D)\bigr),$$0: complete intersections of three quadrics in $$R(X)=\bigoplus_{D\in K} H^0\bigl(X,\mathcal O_X(D)\bigr),$$1 having three nodes.
• $$R(X)=\bigoplus_{D\in K} H^0\bigl(X,\mathcal O_X(D)\bigr),$$2: minimal resolutions of double covers of $$R(X)=\bigoplus_{D\in K} H^0\bigl(X,\mathcal O_X(D)\bigr),$$3 branched along a plane sextic with a double point and two bitangent lines through the point.
• $$R(X)=\bigoplus_{D\in K} H^0\bigl(X,\mathcal O_X(D)\bigr),$$4: minimal resolutions of double covers of $$R(X)=\bigoplus_{D\in K} H^0\bigl(X,\mathcal O_X(D)\bigr),$$5 branched along a plane sextic with three nodes.
• $$R(X)=\bigoplus_{D\in K} H^0\bigl(X,\mathcal O_X(D)\bigr),$$6: smooth quartic surfaces in $$R(X)=\bigoplus_{D\in K} H^0\bigl(X,\mathcal O_X(D)\bigr),$$7 with one hyperplane section that is the union of four lines.
• $$R(X)=\bigoplus_{D\in K} H^0\bigl(X,\mathcal O_X(D)\bigr),$$8 and $$R(X)=\bigoplus_{D\in K} H^0\bigl(X,\mathcal O_X(D)\bigr),$$9: double covers of $K$0 branched along a smooth plane sextic with three 3-tangent lines, distinguished by whether the cover is trivial or non-trivial over their union (Artebani et al., 2020).

These projective models are obtained using Saint-Donat’s theory of linear systems on K3 surfaces. If $K$1 is nef and big, then either $K$2 is base-point free or $K$3 with $K$4 elliptic and $K$5 a $K$6-curve with $K$7. If $K$8 is base-point free and $K$9 is big and nef, then the associated morphism is either $\mathrm{Cl}(X)$0 or birational onto its image, contracting exactly the $\mathrm{Cl}(X)$1-curves orthogonal to $\mathrm{Cl}(X)$2. This is the mechanism behind the repeated appearance of double-plane, double-Hirzebruch, and quartic models (Artebani et al., 2020).

The Cox ring reflects these models in multigraded form. For K3 surfaces, the degrees of a minimal generating set of $\mathrm{Cl}(X)$3 are constrained to be:

1. classes of $\mathrm{Cl}(X)$4-curves;
2. sums of at most three elements of the Hilbert basis of the nef cone;
3. divisors of the form $\mathrm{Cl}(X)$5, where $\mathrm{Cl}(X)$6 are smooth elliptic curves with $\mathrm{Cl}(X)$7 (Artebani et al., 2020).

In some families the paper gives full presentations. For $\mathrm{Cl}(X)$8,

$\mathrm{Cl}(X)$9

where $R(X)$0 correspond to the $R(X)$1 $R(X)$2-curves, $R(X)$3 to the preimage of the positive section on $R(X)$4, and $R(X)$5 to the preimage of the branch curve. For $R(X)$6, the Cox ring is again a hypersurface ring with one quadratic relation expressing the branch equation of the double cover. For $R(X)$7, the Cox ring has $R(X)$8 generators and $R(X)$9 relations explicitly tied to the branch sextic written in the form

$$R(X)=\bigoplus_{D\in K} H^0\bigl(X,\mathcal O_X(D)\bigr)$$00

in $$R(X)=\bigoplus_{D\in K} H^0\bigl(X,\mathcal O_X(D)\bigr)$$01 (Artebani et al., 2020).

This pattern is consistent across the rank-four families: the $$R(X)=\bigoplus_{D\in K} H^0\bigl(X,\mathcal O_X(D)\bigr)$$02-curves govern the effective cone, selected nef divisors govern the main projective models, and the Cox ring packages both kinds of data into a finitely generated multigraded algebra.

4. Hypersurfaces in Mori dream ambient spaces

A major extrinsic construction of Mori dream K3 surfaces comes from hypersurfaces in Mori dream spaces. For a normal irreducible hypersurface $$R(X)=\bigoplus_{D\in K} H^0\bigl(X,\mathcal O_X(D)\bigr)$$03 defined by a homogeneous section $$R(X)=\bigoplus_{D\in K} H^0\bigl(X,\mathcal O_X(D)\bigr)$$04, with $$R(X)=\bigoplus_{D\in K} H^0\bigl(X,\mathcal O_X(D)\bigr)$$05 a Mori dream space and the embedding good, the fundamental criterion is that the induced map

$$R(X)=\bigoplus_{D\in K} H^0\bigl(X,\mathcal O_X(D)\bigr)$$06

gives an isomorphism

$$R(X)=\bigoplus_{D\in K} H^0\bigl(X,\mathcal O_X(D)\bigr)$$07

if and only if three conditions hold: the characteristic preimage $$R(X)=\bigoplus_{D\in K} H^0\bigl(X,\mathcal O_X(D)\bigr)$$08 is big in $$R(X)=\bigoplus_{D\in K} H^0\bigl(X,\mathcal O_X(D)\bigr)$$09, $$R(X)=\bigoplus_{D\in K} H^0\bigl(X,\mathcal O_X(D)\bigr)$$10 is big in $$R(X)=\bigoplus_{D\in K} H^0\bigl(X,\mathcal O_X(D)\bigr)$$11, and $$R(X)=\bigoplus_{D\in K} H^0\bigl(X,\mathcal O_X(D)\bigr)$$12 is an isomorphism (Artebani et al., 2011).

In the ample and spanned situation, this simplifies. If $$R(X)=\bigoplus_{D\in K} H^0\bigl(X,\mathcal O_X(D)\bigr)$$13 is a Mori dream space of dimension at least $$R(X)=\bigoplus_{D\in K} H^0\bigl(X,\mathcal O_X(D)\bigr)$$14, and $$R(X)=\bigoplus_{D\in K} H^0\bigl(X,\mathcal O_X(D)\bigr)$$15 is defined by a general section of an ample base-point-free class, then under the hypotheses recorded in Corollary 2.3 it suffices to check the characteristic-space bigness condition. In the smooth case, a useful criterion is

$$R(X)=\bigoplus_{D\in K} H^0\bigl(X,\mathcal O_X(D)\bigr)$$16

which implies

$$R(X)=\bigoplus_{D\in K} H^0\bigl(X,\mathcal O_X(D)\bigr)$$17

Since $$R(X)=\bigoplus_{D\in K} H^0\bigl(X,\mathcal O_X(D)\bigr)$$18 is finitely generated, the quotient is finitely generated as well, so $$R(X)=\bigoplus_{D\in K} H^0\bigl(X,\mathcal O_X(D)\bigr)$$19 is automatically a Mori dream space (Artebani et al., 2011).

Specializing dimensionally, if $$R(X)=\bigoplus_{D\in K} H^0\bigl(X,\mathcal O_X(D)\bigr)$$20 is a smooth Fano threefold and $$R(X)=\bigoplus_{D\in K} H^0\bigl(X,\mathcal O_X(D)\bigr)$$21 is a smooth anticanonical divisor, then $$R(X)=\bigoplus_{D\in K} H^0\bigl(X,\mathcal O_X(D)\bigr)$$22 is a K3 surface. The paper does not work out the threefold ambient case in detail, but states directly that Theorem 2.1 and Corollary 2.3 apply verbatim in dimension $$R(X)=\bigoplus_{D\in K} H^0\bigl(X,\mathcal O_X(D)\bigr)$$23, with the extra spannedness condition on $$R(X)=\bigoplus_{D\in K} H^0\bigl(X,\mathcal O_X(D)\bigr)$$24. Consequently, if $$R(X)=\bigoplus_{D\in K} H^0\bigl(X,\mathcal O_X(D)\bigr)$$25 is a smooth toric Fano threefold, $$R(X)=\bigoplus_{D\in K} H^0\bigl(X,\mathcal O_X(D)\bigr)$$26 very ample and spanned, and $$R(X)=\bigoplus_{D\in K} H^0\bigl(X,\mathcal O_X(D)\bigr)$$27, then for a smooth anticanonical K3 surface $$R(X)=\bigoplus_{D\in K} H^0\bigl(X,\mathcal O_X(D)\bigr)$$28,

$$R(X)=\bigoplus_{D\in K} H^0\bigl(X,\mathcal O_X(D)\bigr)$$29

so $$R(X)=\bigoplus_{D\in K} H^0\bigl(X,\mathcal O_X(D)\bigr)$$30 is a Mori dream K3 surface (Artebani et al., 2011).

This gives an ambient-space criterion for producing Mori dream K3 surfaces. It complements intrinsic criteria such as finite automorphism group or polyhedral effective cone by replacing them with verifiable conditions on an ambient Mori dream threefold, class-group restriction, and the irrelevant locus in the Cox spectrum. The same paper emphasizes the main failure modes: $$R(X)=\bigoplus_{D\in K} H^0\bigl(X,\mathcal O_X(D)\bigr)$$31 may fail to be an isomorphism, the characteristic preimage may fail to be big, or new divisor classes may appear on the hypersurface (Artebani et al., 2011).

5. Extremal contractions and singular Mori dream K3 surfaces

The smooth theory does not remain unchanged under contractions. A central problem studied in the extremal-contraction paper is whether a non–Mori Dream K3 surface can admit an extremal contraction of a single $$R(X)=\bigoplus_{D\in K} H^0\bigl(X,\mathcal O_X(D)\bigr)$$32-curve to a singular surface that is a Mori Dream Space. The answer is affirmative. An admissible pair $$R(X)=\bigoplus_{D\in K} H^0\bigl(X,\mathcal O_X(D)\bigr)$$33 consists of a K3 surface $$R(X)=\bigoplus_{D\in K} H^0\bigl(X,\mathcal O_X(D)\bigr)$$34 that is not a Mori Dream Space and a singular surface $$R(X)=\bigoplus_{D\in K} H^0\bigl(X,\mathcal O_X(D)\bigr)$$35 obtained by contracting exactly one smooth rational curve $$R(X)=\bigoplus_{D\in K} H^0\bigl(X,\mathcal O_X(D)\bigr)$$36 with $$R(X)=\bigoplus_{D\in K} H^0\bigl(X,\mathcal O_X(D)\bigr)$$37, such that $$R(X)=\bigoplus_{D\in K} H^0\bigl(X,\mathcal O_X(D)\bigr)$$38 is a Mori Dream Space. The singularity is a single $$R(X)=\bigoplus_{D\in K} H^0\bigl(X,\mathcal O_X(D)\bigr)$$39 rational double point (Garbagnati, 2015).

Let

$$R(X)=\bigoplus_{D\in K} H^0\bigl(X,\mathcal O_X(D)\bigr)$$40

Then there exists a K3 surface $$R(X)=\bigoplus_{D\in K} H^0\bigl(X,\mathcal O_X(D)\bigr)$$41 with $$R(X)=\bigoplus_{D\in K} H^0\bigl(X,\mathcal O_X(D)\bigr)$$42, and $$R(X)=\bigoplus_{D\in K} H^0\bigl(X,\mathcal O_X(D)\bigr)$$43. Under the hypothesis that there is no $$R(X)=\bigoplus_{D\in K} H^0\bigl(X,\mathcal O_X(D)\bigr)$$44-curve $$R(X)=\bigoplus_{D\in K} H^0\bigl(X,\mathcal O_X(D)\bigr)$$45 with $$R(X)=\bigoplus_{D\in K} H^0\bigl(X,\mathcal O_X(D)\bigr)$$46, the nef cones coincide: $$R(X)=\bigoplus_{D\in K} H^0\bigl(X,\mathcal O_X(D)\bigr)$$47 From this one obtains the central criterion: under the same hypothesis, $$R(X)=\bigoplus_{D\in K} H^0\bigl(X,\mathcal O_X(D)\bigr)$$48 is a Mori Dream Space if and only if $$R(X)=\bigoplus_{D\in K} H^0\bigl(X,\mathcal O_X(D)\bigr)$$49 is a Mori Dream Space (Garbagnati, 2015).

This leads to a lattice-theoretic classification of admissible configurations. The paper proves that if $$R(X)=\bigoplus_{D\in K} H^0\bigl(X,\mathcal O_X(D)\bigr)$$50 is an admissible pair, then $$R(X)=\bigoplus_{D\in K} H^0\bigl(X,\mathcal O_X(D)\bigr)$$51; under the no-intersection-$$R(X)=\bigoplus_{D\in K} H^0\bigl(X,\mathcal O_X(D)\bigr)$$52 hypothesis, one has $$R(X)=\bigoplus_{D\in K} H^0\bigl(X,\mathcal O_X(D)\bigr)$$53; if $$R(X)=\bigoplus_{D\in K} H^0\bigl(X,\mathcal O_X(D)\bigr)$$54, only finitely many Néron–Severi lattices occur; and for $$R(X)=\bigoplus_{D\in K} H^0\bigl(X,\mathcal O_X(D)\bigr)$$55 a complete list is given. It also proves that $$R(X)=\bigoplus_{D\in K} H^0\bigl(X,\mathcal O_X(D)\bigr)$$56 is the minimal Picard number for such a phenomenon and constructs infinitely many examples with $$R(X)=\bigoplus_{D\in K} H^0\bigl(X,\mathcal O_X(D)\bigr)$$57 (Garbagnati, 2015).

The later singular-K3 paper reframes this in more general terms. A singular K3 surface is a normal projective surface with at most rational double points, $$R(X)=\bigoplus_{D\in K} H^0\bigl(X,\mathcal O_X(D)\bigr)$$58, and $$R(X)=\bigoplus_{D\in K} H^0\bigl(X,\mathcal O_X(D)\bigr)$$59. If $$R(X)=\bigoplus_{D\in K} H^0\bigl(X,\mathcal O_X(D)\bigr)$$60 is the minimal resolution, then $$R(X)=\bigoplus_{D\in K} H^0\bigl(X,\mathcal O_X(D)\bigr)$$61 is a smooth K3 surface and the exceptional curves form ADE Dynkin diagrams. The main theorem states: if the Picard lattice $$R(X)=\bigoplus_{D\in K} H^0\bigl(X,\mathcal O_X(D)\bigr)$$62 of a singular K3 surface is a Mori dream lattice, then $$R(X)=\bigoplus_{D\in K} H^0\bigl(X,\mathcal O_X(D)\bigr)$$63 is a Mori dream surface (Laface et al., 2024).

For Picard rank $$R(X)=\bigoplus_{D\in K} H^0\bigl(X,\mathcal O_X(D)\bigr)$$64, the same paper gives a sharp criterion: $$R(X)=\bigoplus_{D\in K} H^0\bigl(X,\mathcal O_X(D)\bigr)$$65 It then applies this to a K3 surface with a single $$R(X)=\bigoplus_{D\in K} H^0\bigl(X,\mathcal O_X(D)\bigr)$$66 singularity. If $$R(X)=\bigoplus_{D\in K} H^0\bigl(X,\mathcal O_X(D)\bigr)$$67 has Picard rank $$R(X)=\bigoplus_{D\in K} H^0\bigl(X,\mathcal O_X(D)\bigr)$$68, its singular locus consists of a single point of type $$R(X)=\bigoplus_{D\in K} H^0\bigl(X,\mathcal O_X(D)\bigr)$$69, and $$R(X)=\bigoplus_{D\in K} H^0\bigl(X,\mathcal O_X(D)\bigr)$$70 contains an irreducible curve of negative self-intersection, then $$R(X)=\bigoplus_{D\in K} H^0\bigl(X,\mathcal O_X(D)\bigr)$$71 is a Mori dream space provided

$$R(X)=\bigoplus_{D\in K} H^0\bigl(X,\mathcal O_X(D)\bigr)$$72

The proof uses the inverse Cartan matrix of $$R(X)=\bigoplus_{D\in K} H^0\bigl(X,\mathcal O_X(D)\bigr)$$73, the decomposition $$R(X)=\bigoplus_{D\in K} H^0\bigl(X,\mathcal O_X(D)\bigr)$$74, and the effectivity criterion

$$R(X)=\bigoplus_{D\in K} H^0\bigl(X,\mathcal O_X(D)\bigr)$$75

for the fractional exceptional part $$R(X)=\bigoplus_{D\in K} H^0\bigl(X,\mathcal O_X(D)\bigr)$$76 (Laface et al., 2024).

These results show that singularities can improve Mori dream behavior. The weighted-projective examples in the same work make this concrete: very general hypersurfaces such as $$R(X)=\bigoplus_{D\in K} H^0\bigl(X,\mathcal O_X(D)\bigr)$$77, $$R(X)=\bigoplus_{D\in K} H^0\bigl(X,\mathcal O_X(D)\bigr)$$78, $$R(X)=\bigoplus_{D\in K} H^0\bigl(X,\mathcal O_X(D)\bigr)$$79, $$R(X)=\bigoplus_{D\in K} H^0\bigl(X,\mathcal O_X(D)\bigr)$$80, and $$R(X)=\bigoplus_{D\in K} H^0\bigl(X,\mathcal O_X(D)\bigr)$$81 carry explicit negative curves with positive pairwise intersection, hence are Mori dream by the rank-two criterion (Laface et al., 2024).

6. Moduli, coverings, and higher-dimensional analogues

In the Picard-rank-four classification, the explicit models are strong enough to yield unirationality results for the associated lattice-polarized moduli spaces. The moduli spaces $$R(X)=\bigoplus_{D\in K} H^0\bigl(X,\mathcal O_X(D)\bigr)$$82 are unirational for

$$R(X)=\bigoplus_{D\in K} H^0\bigl(X,\mathcal O_X(D)\bigr)$$83

and $$R(X)=\bigoplus_{D\in K} H^0\bigl(X,\mathcal O_X(D)\bigr)$$84 is rational. The proofs proceed by describing the relevant branch sextics, quartics, or nets of quadrics as open subsets of linear systems or products of vector spaces, then quotienting by the corresponding projective automorphism group (Artebani et al., 2020).

A distinct but related perspective comes from étale coverings in codimension $$R(X)=\bigoplus_{D\in K} H^0\bigl(X,\mathcal O_X(D)\bigr)$$85. For a Mori Dream Enriques surface $$R(X)=\bigoplus_{D\in K} H^0\bigl(X,\mathcal O_X(D)\bigr)$$86, the canonical $$R(X)=\bigoplus_{D\in K} H^0\bigl(X,\mathcal O_X(D)\bigr)$$87-covering $$R(X)=\bigoplus_{D\in K} H^0\bigl(X,\mathcal O_X(D)\bigr)$$88 constructed from the Cox-ring and toric framework coincides with the universal topological covering, and $$R(X)=\bigoplus_{D\in K} H^0\bigl(X,\mathcal O_X(D)\bigr)$$89 is a K3 surface. However, the universal K3 covering of an Enriques surface has infinite automorphism group, so it is never a Mori Dream Space. At the same time, it admits a canonical embedding into a $$R(X)=\bigoplus_{D\in K} H^0\bigl(X,\mathcal O_X(D)\bigr)$$90-factorial toric variety with free class group, via the universal $$R(X)=\bigoplus_{D\in K} H^0\bigl(X,\mathcal O_X(D)\bigr)$$91-cover of the canonical ambient toric variety (Rossi, 2019). This sharply distinguishes “Mori dream K3 surface” from “K3 surface arising as a canonical cover of a Mori dream surface.”

Finally, the higher-dimensional hyperkähler analogue replaces the K3 Picard lattice by Markman’s extended Hodge lattice. For projective irreducible holomorphic symplectic manifolds of K3$$R(X)=\bigoplus_{D\in K} H^0\bigl(X,\mathcal O_X(D)\bigr)$$92-type, the Mori cone is generated by the positive cone together with images $$R(X)=\bigoplus_{D\in K} H^0\bigl(X,\mathcal O_X(D)\bigr)$$93 of algebraic classes $$R(X)=\bigoplus_{D\in K} H^0\bigl(X,\mathcal O_X(D)\bigr)$$94 satisfying

$$R(X)=\bigoplus_{D\in K} H^0\bigl(X,\mathcal O_X(D)\bigr)$$95

Extremal rays satisfy the uniform bound

$$R(X)=\bigoplus_{D\in K} H^0\bigl(X,\mathcal O_X(D)\bigr)$$96

Although the paper does not assert that every projective K3$$R(X)=\bigoplus_{D\in K} H^0\bigl(X,\mathcal O_X(D)\bigr)$$97-type manifold is a Mori Dream Space in Hu–Keel’s sense, it describes a wall-and-chamber structure governed by a Weyl group generated by reflections in exceptional divisors, with behavior explicitly compared to Mori dream geometry (Bayer et al., 2013).

This suggests a precise higher-dimensional analogue of the surface picture: on K3 surfaces, the nef and effective cones are controlled by the Néron–Severi lattice and its $$R(X)=\bigoplus_{D\in K} H^0\bigl(X,\mathcal O_X(D)\bigr)$$98-classes; on K3$$R(X)=\bigoplus_{D\in K} H^0\bigl(X,\mathcal O_X(D)\bigr)$$99-type manifolds, the corresponding cone structure is controlled by the extended lattice and classes of square at least $\mathbb C$00, together with the constraint involving the primitive vector $\mathbb C$01 (Bayer et al., 2013).

A Mori dream K3 surface is therefore best viewed not as an isolated class of surfaces but as a meeting point of several equivalent structures: finite automorphism group, polyhedral effective cone, finitely generated Cox ring, and a Néron–Severi lattice of restricted reflective type. Smooth examples admit explicit realizations as quartics, double planes, double covers of Hirzebruch surfaces, or hypersurfaces in Mori dream ambient spaces; singular examples arise both from extremal contractions and from weighted-projective models; and the surrounding theory extends naturally to Enriques covers and to K3$\mathbb C$02-type hyperkähler geometry (Artebani et al., 2020, Artebani et al., 2011, Garbagnati, 2015, Laface et al., 2024, Rossi, 2019, Bayer et al., 2013).