Additive Action in Algebraic Geometry

• Additive action is a regular effective action of the vector group Gₐⁿ on an irreducible variety where the open orbit is isomorphic to affine space Aⁿ.
• It serves as the unipotent counterpart to toric actions, analyzed via birational conjugacy, Demazure roots, and lattice polytope criteria.
• The theory connects local algebra concepts, invariant multilinear forms, and computational criteria to classify actions on projective spaces and hypersurfaces.

An additive action on an irreducible algebraic variety $X$ of dimension $n$ is a regular effective action

$\mathbb G_a^n\times X\to X$

with an open orbit, where $\mathbb G_a$ is the additive group of the ground field and $\mathbb G_a^n$ is the $n$-dimensional vector group. Equivalently, $X$ is an equivariant compactification of affine space $\mathbb A^n$: the open orbit is isomorphic to $\mathbb A^n$, and the complement is the boundary of the compactification. In contemporary algebraic geometry, additive actions are studied as the unipotent counterpart of torus actions in toric geometry, and their structure is analyzed through birational geometry, toric combinatorics, local finite-dimensional algebras, locally nilpotent derivations, and explicit projective embeddings (Arzhantsev, 2023).

1. Definition and conceptual position

The standard definition fixes three features simultaneously: the acting group is the commutative unipotent group $\mathbb G_a^n$, the action is effective, and the orbit is open in the Zariski topology. In dimension $n$0, this makes the open orbit a dense copy of $n$1, so additive actions are a distinguished form of equivariant compactification of affine space. Several works emphasize that this notion plays, for “additive geometry,” a role analogous to the role of a dense torus orbit in toric geometry, but with a vector group replacing the torus (Arzhantsev et al., 2015).

Unlike toric actions, additive actions on a fixed variety are often not unique. This non-uniqueness is visible already on projective space and on toric surfaces, and it motivates classification problems phrased either up to isomorphism, up to equivalence in projective space, or up to conjugacy inside a larger transformation group such as $n$2, $n$3, or the affine Cremona group (Dzhunusov, 2020).

For projective subvarieties $n$4, one often distinguishes induced additive actions: regular effective actions on $n$5 with an open orbit that extend to a regular action on the ambient projective space. In the hypersurface setting this extension is central, because it permits a direct translation between geometry and finite-dimensional local algebra data (Beldiev, 2023).

A recurrent structural principle is that additive actions are encoded infinitesimally by locally nilpotent operators. In affine and relative settings, a $n$6-action on $n$7 is described by a locally finite iterative higher derivation $n$8, equivalently by a comorphism

$n$9

and this yields a canonical filtration, associated graded algebra, and relative Rees algebra attached to the action (Dubouloz et al., 2019).

2. Conjugacy, birational uniqueness, and ambient transformation groups

A general birational rigidity statement holds for additive actions with open orbit. Any two additive actions on the same irreducible variety $\mathbb G_a^n\times X\to X$0 are conjugate in $\mathbb G_a^n\times X\to X$1; more precisely, their open orbits are isomorphic as varieties, and the induced identification gives a birational automorphism sending one action to the other. If the two actions have the same open orbit $\mathbb G_a^n\times X\to X$2, then the conjugating map lies in $\mathbb G_a^n\times X\to X$3 (Arzhantsev, 2023).

When $\mathbb G_a^n\times X\to X$4, the group $\mathbb G_a^n\times X\to X$5 is the affine Cremona group

$\mathbb G_a^n\times X\to X$6

that is, the group of polynomial automorphisms of affine $\mathbb G_a^n\times X\to X$7-space. This is a stronger statement than abstract birational conjugacy: it locates the conjugating element inside a concrete polynomial automorphism group rather than merely in the full Cremona group

$\mathbb G_a^n\times X\to X$8

In the special case $\mathbb G_a^n\times X\to X$9, the conjugating polynomial automorphism can be written explicitly in terms of the corresponding local algebra (Arzhantsev, 2023).

On complete toric varieties, the conjugacy problem is sharpened from birational to biregular classification. A complete toric variety admits an additive action if and only if it admits one normalized by the acting torus, and normalized additive actions are classified by complete collections of Demazure roots of the fan. Moreover, any two normalized additive actions on a toric variety are isomorphic (Arzhantsev et al., 2015).

Uniqueness, however, is not automatic even in the complete toric case. A complete toric variety can carry more than one additive action up to isomorphism, and the precise uniqueness criterion is combinatorial. For such a variety $\mathbb G_a$0, every additive action is isomorphic to the normalized one if and only if for each basis ray $\mathbb G_a$1 the corresponding set of Demazure roots satisfies

$\mathbb G_a$2

equivalently, the preorder on the basis rays is trivial, or equivalently,

$\mathbb G_a$3

for a maximal unipotent subgroup $\mathbb G_a$4 (Dzhunusov, 2020).

3. Toric varieties: fans, Demazure roots, and polytope criteria

For toric varieties, additive actions admit a fan-theoretic and polyhedral description. If $\mathbb G_a$5 is a toric variety with fan $\mathbb G_a$6, then a $\mathbb G_a$7-action normalized by the acting torus corresponds to a Demazure root $\mathbb G_a$8, where for a distinguished ray $\mathbb G_a$9 with primitive generator $\mathbb G_a^n$0,

$\mathbb G_a^n$1

A normalized additive action corresponds to a complete collection of Demazure roots $\mathbb G_a^n$2 satisfying

$\mathbb G_a^n$3

For complete toric varieties, the existence of an additive action is equivalent to the existence of such a complete collection (Arzhantsev et al., 2015).

This criterion has an equivalent geometric formulation in terms of the fan. A complete toric variety admits an additive action if and only if the rays can be ordered so that the first $\mathbb G_a^n$4 primitive generators form a basis of $\mathbb G_a^n$5 and all remaining rays lie in the negative octant with respect to this basis (Dzhunusov, 2019). In computational work, this condition is refined further: it is enough to inspect maximal cones, because the basis rays can be chosen as the extremal rays of a maximal cone, and uniqueness is then tested by checking whether the corresponding Demazure root sets are singletons (Levicán et al., 4 Nov 2025).

For projective toric varieties, the same phenomenon is encoded by the defining lattice polytope. If $\mathbb G_a^n$6 is a very ample lattice polytope, then the associated projective toric variety $\mathbb G_a^n$7 admits an additive action if and only if $\mathbb G_a^n$8 is inscribed in a rectangle: there exists a vertex $\mathbb G_a^n$9 such that the primitive edge directions at $n$0 form a lattice basis, and every facet not passing through $n$1 has inward normal satisfying the required sign condition in that basis (Shafarevich, 2021).

This criterion becomes especially rigid for projective toric hypersurfaces. An $n$2-dimensional projective toric variety is a hypersurface in projective space exactly when its defining polytope has $n$3 lattice points. Among such hypersurfaces, the only ones admitting additive actions are

$n$4

and for $n$5,

$n$6

that is, projective space and quadrics of rank $n$7 and rank $n$8. The classification is obtained by showing that, up to translation and lattice basis change, the only full-dimensional very ample lattice polytopes with $n$9 lattice points that are inscribed in a rectangle are

$X$0

with associated quadrics

$X$1

respectively (Shafarevich, 2020).

A further geometric reinterpretation identifies additive toric geometry with Euler-symmetry. For projective toric varieties, the following are equivalent: Euler-symmetry with respect to some nondegenerate embedding, Euler-symmetry with respect to any nondegenerate linearly normal embedding, and the existence of an additive action. In this setting, Euler points are exactly the smooth points lying in the $X$2-orbit of a torus-fixed point (Shafarevich, 2021).

4. Projective space and the Hassett–Tschinkel correspondence

The prototype of additive geometry is projective space. Additive actions on $X$3 are classified by local commutative associative unital algebras $X$4 of dimension $X$5. If $X$6 is the maximal ideal of $X$7, then after choosing a basis $X$8 of $X$9, the action on $\mathbb A^n$0 is given by multiplication by

$\mathbb A^n$1

and the open orbit is the affine chart $\mathbb A^n$2 (Arzhantsev, 2023).

Writing

$\mathbb A^n$3

produces polynomials $\mathbb A^n$4, called basic polynomials. In a suitable basis they are triangular,

$\mathbb A^n$5

so the map $\mathbb A^n$6 is a polynomial automorphism of $\mathbb A^n$7, hence an element of the affine Cremona group. The main theorem of the conjugacy theory for $\mathbb A^n$8 states that this automorphism conjugates the additive action corresponding to $\mathbb A^n$9 to the standard additive action on projective space (Arzhantsev, 2023).

The proof is explicit. One considers the automorphism

$\mathbb A^n$0

with inverse

$\mathbb A^n$1

If $\mathbb A^n$2 and $\mathbb A^n$3, the standard action is translation

$\mathbb A^n$4

whereas the action coming from $\mathbb A^n$5 is multiplication by $\mathbb A^n$6. The identity

$\mathbb A^n$7

then implies

$\mathbb A^n$8

so $\mathbb A^n$9 is the required conjugating polynomial automorphism (Arzhantsev, 2023).

This correspondence also shows that subgroup conjugacy in the affine Cremona group is finer than conjugacy inside $\mathbb G_a^n$0. A single conjugacy class of $\mathbb G_a^n$1-subgroups in the affine Cremona group may intersect $\mathbb G_a^n$2 in several distinct $\mathbb G_a^n$3-conjugacy classes, and the basic polynomials identify the specific Cremona transformation relating a given action to the standard one (Arzhantsev, 2023).

A stronger rigidity appears when one imposes a boundary condition on one-parameter subgroups. For additive actions on $\mathbb G_a^n$4, there is exactly one action satisfying the OP-condition for each $\mathbb G_a^n$5, namely the standard action

$\mathbb G_a^n$6

corresponding to the local algebra with square-zero maximal ideal

$\mathbb G_a^n$7

(Shafarevich, 6 May 2025).

5. Hypersurfaces, $\mathbb G_a^n$8-pairs, Gorenstein algebras, and invariant forms

For projective hypersurfaces, the classification problem is governed by an algebra–geometry correspondence extending the projective-space case. Induced additive actions on hypersurfaces in $\mathbb G_a^n$9 not contained in any hyperplane are in one-to-one correspondence with pairs $n$00, where $n$01 is a local commutative associative unital algebra of dimension $n$02, $n$03 is its maximal ideal, and $n$04 is a generating hyperplane. Such pairs are called $n$05-pairs (Beldiev, 2023).

Given $n$06, the hypersurface is obtained as

$n$07

or equivalently, in the notation used elsewhere,

$n$08

Its degree is determined by the nilpotent filtration: $n$09 The defining equation can be written using the logarithm map: $n$10 with $n$11 the canonical projection (Borovik et al., 2024).

For hypersurfaces of degree $n$12, the correspondence sharpens to a Gorenstein condition. Induced additive actions on hypersurfaces of degree $n$13 in $n$14 correspond to pairs $n$15 where $n$16 is a Gorenstein local algebra of dimension $n$17 with

$n$18

and $n$19 is a hyperplane complementary to $n$20 (Beldiev, 2023). This algebraic description controls both the geometry of the hypersurface and the uniqueness of the action.

A central theorem states that an induced additive action on a projective hypersurface is unique if and only if the hypersurface is non-degenerate, equivalently not a projective cone. In algebraic terms, uniqueness is equivalent to the condition that the corresponding $n$21-pair has $n$22 Gorenstein and $n$23 complementary to the socle (Beldiev, 2023). Conversely, degeneracy forces non-uniqueness, because one can reduce along ideals contained in $n$24 and recover distinct $n$25-pairs producing the same cone (Beldiev, 2023).

An alternative but equivalent formalism replaces the pair $n$26 by an irreducible invariant symmetric multilinear form. For hypersurfaces of degree $n$27, additive actions are in natural bijection with pairs $n$28, where $n$29 is a local algebra of dimension $n$30 and $n$31 is an irreducible invariant $n$32-linear symmetric form, defined up to scalar (Arzhantsev et al., 2013). In the quadratic case this formalism yields explicit classifications: a non-degenerate quadric admits a unique additive action up to equivalence, while quadrics of corank one admit richer families classified by canonical forms of symmetric matrices (Arzhantsev et al., 2013).

The hypersurface theory has several sharp classification results. A cubic hypersurface admits an additive structure if and only if, after a suitable choice of homogeneous coordinates, its defining equation has the form

$n$33

where $n$34 is a non-degenerate cubic form in $n$35 variables; the additive structure is unique if and only if the cubic is non-degenerate in $n$36 variables, equivalently $n$37 (Bazhov, 2013).

High-degree hypersurfaces are also rigid. If $n$38 admits an induced additive action, then $n$39, and there is a unique such hypersurface of degree $n$40, corresponding to

$n$41

For degree $n$42, there is exactly one non-degenerate hypersurface for every $n$43, corresponding to

$n$44

and the degrees $n$45 and $n$46 admit explicit finite classifications by Gorenstein local algebras, with an infinite family appearing in degree $n$47 for $n$48 (Beldiev, 13 Apr 2025).

6. Orbit structure, boundary geometry, normality, and explicit restrictions

The boundary $n$49 of the open orbit often carries decisive information. For a projective hypersurface $n$50 coming from an $n$51-pair $n$52, the open orbit is the locus $n$53, while the boundary lies in the hyperplane $n$54 and is defined by the top homogeneous term $n$55. This boundary hypersurface is always degenerate; in particular, it is either a hyperplane or a singular hypersurface (Arzhantsev et al., 2024).

Normality can be read off directly from the first two highest-degree pieces of the defining equation. If

$n$56

then the hypersurface is normal if and only if $n$57 and $n$58 are coprime. A direct consequence is that $n$59 is normal whenever $n$60 is square-free (Arzhantsev et al., 2024).

The boundary geometry is flexible in another direction: for any non-degenerate projective hypersurface $n$61, there exist an integer $n$62 and a non-degenerate hypersurface $n$63 with an induced additive action such that the boundary of the open orbit in $n$64 is a projective cone over $n$65 (Arzhantsev et al., 2024). This shows that additive compactifications can be constructed with essentially arbitrary prescribed boundary hypersurface, up to coning.

Several classification problems refine the orbit decomposition. For induced additive actions on projective hypersurfaces with finitely many orbits, the corresponding $n$66-pairs are exactly

$n$67

and

$n$68

Geometrically this yields only certain quadrics and closely related low-degree hypersurfaces (Borovik et al., 2024).

A different restriction is the OP-condition for additive actions on projective hypersurfaces: every boundary orbit must be reachable as a limit of a one-parameter subgroup from a point in the open orbit. In the non-degenerate case this forces the degree to be

$n$69

The complete classification then consists of quadrics of the form

$n$70

and the cubic hypersurface

$n$71

together with their natural degenerate extensions in larger projective spaces (Shafarevich, 6 May 2025).

Orbit finiteness and non-uniqueness also appear on surfaces. Projective surfaces with du Val singularities admitting an additive action with a finite number of orbits form an explicit list built from $n$72, $n$73, Hirzebruch surfaces, and controlled blowups of boundary points; singular examples are obtained by contracting $n$74-curves. In this setting, some surfaces carry $n$75-parameter families of pairwise non-isomorphic additive actions, answering a question of Hassett and Tschinkel (Perepechko, 11 Aug 2025).

The toric-surface case is especially concrete. A complete toric surface admitting an additive action has either exactly one additive action up to isomorphism, when the fan is wide, or exactly two non-isomorphic additive actions, one normalized and one non-normalized (Dzhunusov, 2019).

7. Relative, affine, and computational perspectives

Beyond projective classification, additive actions are studied through canonical algebraic invariants. For a $n$76-action on an affine $n$77-scheme $n$78, the induced filtration

$n$79

defines the relative Rees algebra

$n$80

and the associated graded algebra

$n$81

The Rees algebra simultaneously yields a deformation from $n$82 to the associated graded degeneration and a canonical equivariant completion

$n$83

(Dubouloz et al., 2019).

In the torsor case, this construction reduces to a projective-bundle model: if $n$84 is a $n$85-torsor, then $n$86, and $n$87 is the projective bundle $n$88, with $n$89 the complement of a distinguished section (Dubouloz et al., 2019). This places additive actions within the broader framework of filtrations, degenerations, affine modifications, and finite-generation criteria.

Computationally, additive toric geometry has been implemented in the Macaulay2 package AdditiveToricVarieties, which provides methods based on Demazure roots and complete-collection criteria to decide whether a complete toric variety is additive and whether the additive action is unique (Levicán et al., 4 Nov 2025). The package is applied to smooth Fano toric varieties up to dimension $n$90, and one structural consequence proved in that work is that every smooth complete toric variety of Picard rank $n$91 is additive, while uniqueness in that class occurs only for

$n$92

(Levicán et al., 4 Nov 2025).

A broader implication of these developments is that additive action theory is now organized around several mutually compatible languages: birational conjugacy and Cremona transformations, toric fans and lattice polytopes, local algebras and socles, invariant multilinear forms, locally nilpotent derivations and higher derivations, and explicit computational criteria. This suggests that the modern theory is less a single classification problem than an interface between unipotent group actions, compactifications of affine space, and the internal algebraic structure of the varieties that carry them.