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**Toric symmetry of \(\mathbb{CP}^3\).**
*(English)*
Zbl 1284.14075

Toric symmetries of a toric variety are the automorphisms of the lattice \(\mathbb{Z}^n\) that permute the cones of its fan. The authors are interested in symmetries that act non–trivially on Chow homology, and Gromov–Witten and Donaldson–Thomas invariants. Non–trivial toric symmetries provide a new way of computing some of the invariants, as an alternative to fixed point localization, topological vertex and the remodeling conjecture. It manifests at the level of invariants themselves, rather than generating functions, and is difficult to detect by other techniques.

In this paper a case study of toric symmetries is undertaken for the toric blowups of \(\mathbb{P}^3\). The latter are obtained by iterated blowup of \(\mathbb{P}^3\) along toric invariant subvarieties only, so blowups with the centers in the exceptional locus, for example, are excluded. The prototypical example is the Cremona symmetry of \(\mathbb{P}^3\) induced by the resolution of the birational Cremona transformation.

The paper identifies four classes of toric blowups that have non–trivial toric symmetries (up to relabeling of cones); three classes have a unique symmetry, and one has exactly three distinct symmetries. Explicit descriptions of the four classes and their symmetries are given, in particular their toric polytopes are identified. They include permutohedron, cyclohedron and other graph associahedra. It is mentioned that in the case of \(\mathbb{P}^2\) the only toric blowup with non–trivial symmetry is the blowup of \(\mathbb{P}^2\) at the three torus fixed points, and the symmetry is the two dimensional Cremona symmetry.

In this paper a case study of toric symmetries is undertaken for the toric blowups of \(\mathbb{P}^3\). The latter are obtained by iterated blowup of \(\mathbb{P}^3\) along toric invariant subvarieties only, so blowups with the centers in the exceptional locus, for example, are excluded. The prototypical example is the Cremona symmetry of \(\mathbb{P}^3\) induced by the resolution of the birational Cremona transformation.

The paper identifies four classes of toric blowups that have non–trivial toric symmetries (up to relabeling of cones); three classes have a unique symmetry, and one has exactly three distinct symmetries. Explicit descriptions of the four classes and their symmetries are given, in particular their toric polytopes are identified. They include permutohedron, cyclohedron and other graph associahedra. It is mentioned that in the case of \(\mathbb{P}^2\) the only toric blowup with non–trivial symmetry is the blowup of \(\mathbb{P}^2\) at the three torus fixed points, and the symmetry is the two dimensional Cremona symmetry.

Reviewer: Sergiy Koshkin (Houston)

### MSC:

14N35 | Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects) |

14M25 | Toric varieties, Newton polyhedra, Okounkov bodies |

52B20 | Lattice polytopes in convex geometry (including relations with commutative algebra and algebraic geometry) |