On geometry of binary symmetric models of phylogenetic trees. (English) Zbl 1147.14027

Buczyńska, Weronika; Wiśniewski, Jarosław A.

J. Eur. Math. Soc. (JEMS) 9, No. 3, 609-635 (2007).

The basic object studied in the present paper is a trivalent tree, i.e.an unoriented graph \(\mathcal T\) without cycles whose vertices have either valency one (the “leaves”) or three (the inner “nodes”). It gives rise to a certain lattice polytope \(\Delta(\mathcal T)\) with vertices corresponding to networks of mutually non-intersecting paths connecting leaves along edges of \(\mathcal T\). In the easiest case of three leaves being connected to a unique node, \(\Delta(\mathcal T)\) is the three-dimensional standard tetrahedron. In general, if \(\ell\) denotes the number of leaves, \(\Delta(\mathcal T)\) has exactly \(2^{\ell-1}\) vertices.
The first result in the present paper is that the 4th scalar multiple of \(\Delta(\mathcal T)\) is a reflexive polytope, i.e.the associated toric variety \(X(\mathcal T)\) is Fano of index \(4\) having at most Gorenstein terminal singularities. Moreover, the authors show (by providing an explicit polyhedral subdivision of \(\partial\Delta(\mathcal T)\)) that \(X(\mathcal T)\) admits a small, crepant resolution.
While the authors give the example that different trees (with the same number of leaves) might lead to mutually non-isomorphic polytopes, it was their striking observation that the Ehrhart polynomial of \(\Delta(\mathcal T)\) (i.e.the Hilbert polynomial of \(X(\mathcal T)\)) does depend only on \(\ell\). The second result of the paper is an explanation (and proof) of this fact by presenting a flat family that contains all varieties \(X(\mathcal T)\) with a fixed number \(\ell\) as special fibers. In particular, the freedom of choosing the easiest tree for a given \(\ell\) leads to a formula for the Ehrhart polynomial of \(\Delta(\mathcal T)\).

Reviewer: Klaus Altmann (Berlin)

MSC:

14M25	Toric varieties, Newton polyhedra, Okounkov bodies
14C05	Parametrization (Chow and Hilbert schemes)
14J45	Fano varieties
92B10	Taxonomy, cladistics, statistics in mathematical biology

Keywords:

toric variety; flat family; resolution; phylogenetic tree; reflexive polytope

Software:

polymake

References:

[1]	Allman, E. S., A. Rhodes, J.: The identifiability of tree topology for phylogenetic mod- els, including covarion and mixture models. J. Comput. Biol. 13 , 1101-1113 (2006)
[2]	Altmann, K.: Deformation Theory. Notes available at http://page.mi.fu-berlin.de/ altmann/PAPER/dmv.ps (1995)
[3]	Białynicki-Birula, A.: Quotients by actions of groups. In: Encyclopedia Math. Sci. 131, Springer, 1-82 (2003) · Zbl 1061.14046
[4]	Billera, L., Holmes, S. P., Vogtmann, K.: Geometry of the space of phylogenetic trees. Adv. Appl. Math. 27 , 733-767 (2001) · Zbl 0995.92035 · doi:10.1006/aama.2001.0759
[5]	Bruns, W., Gubeladze, J., Trung, N. V.: Normal polytopes, triangulations and Koszul al- gebras. J. Reine Angew. Math. 485 , 123-160 (1997) · Zbl 0866.20050
[6]	Buczyńska, W., Wiśniewski, J. A.: On phylogenetic trees-a geometer’s view. math.AG/0601357 · Zbl 1147.14027
[7]	Casanellas, M., Garcia, L. D., Sullivant, S.: Catalog of small trees. In: Algebraic Statistics for Computational Biology, L. Pachter and B. Sturmfels (eds.), Cambridge Univ. Press, 291-304 (2005) · Zbl 1376.62075
[8]	Eisenbud, D.: Commutative Algebra with a View Toward Algebraic Geometry. Springer (1995). · Zbl 0819.13001
[9]	Eisenbud, D., Sturmfels, B.: Binomial ideals. Duke Math. J. 84 , 1-45 (1996) · Zbl 0873.13021 · doi:10.1215/S0012-7094-96-08401-X
[10]	Eriksson, N., Ranestad, K., Sturmfels, B., Sullivant, S.: Phylogenetic algebraic geom- etry. In: Projective Varieties with Unexpected Properties, de Gruyter, 237-255 (2005) · Zbl 1106.14050
[11]	Evans, S., Speed, T.: Invariants of some probability models used in phylogenetic infer- ence. Ann. Statist. 21 , 355-377 (1993) · Zbl 0772.92012 · doi:10.1214/aos/1176349030
[12]	Felsenstein, J.: Inferring Phylogenies. Sinauer Associates (2003)
[13]	Fulton, W.: Introduction to Toric Varieties. Princeton Univ. Press (1993) · Zbl 0813.14039
[14]	Hartshorne, R.: Algebraic Geometry. Grad. Texts in Math. 52, Springer (1977) · Zbl 0367.14001
[15]	Kollár, J., Mori, Sh.: Birational Geometry of Algebraic Varieties. Cambridge Univ. Press (1998) · Zbl 0926.14003
[16]	Mumford, D., Fogarty, J.: Geometric Invariant Theory. 2nd ed., Springer (1982) · Zbl 0504.14008
[17]	Namikawa, Y.: Smoothing Fano 3-folds. J. Algebraic Geom. 6 , 307-324 (1997) · Zbl 0906.14019
[18]	Oda, T.: Convex Bodies and Algebraic Geometry. Springer (1988) · Zbl 0628.52002
[19]	Pachter, L., Sturmfels, B. (eds.): Algebraic Statistics for Computational Biology. Cam- bridge Univ. Press (2005) · Zbl 1108.62118 · doi:10.1017/CBO9780511610684
[20]	Reid, M.: What is a flip?. Colloquium talk notes, Salt Lake City (1992), http://www.maths.warwick.ac.uk/ miles/3folds/what_flip.pdf
[21]	Semple, C., Steel, M.: Phylogenetics. Oxford Univ. Press (2003) · Zbl 1043.92026
[22]	Sturmfels, B.: Equations defining toric varieties. In: Algebraic Geometry (Santa Cruz, 1995), Proc. Sympos. Pure Math. 62, Amer. Math. Soc., 437-449 (1997). · Zbl 0914.14022
[23]	Sturmfels, B.: Gröbner Bases and Convex Polytopes. Univ. Lecture Ser. 8, Amer. Math. Soc. (1996) · Zbl 0856.13020
[24]	Sturmfels, B., Sullivant, S.: Toric ideals of phylogenetic invariants. J. Comput. Biol. 12 , 204-228 (2005) · Zbl 1391.13058
[25]	Szekely, L., Steel, M., Erd\Acute\Acute os, P.: Fourier calculus on evolutionary trees. Adv. Appl. Math. 14 , 200-216 (1993) · Zbl 0794.05014 · doi:10.1006/aama.1993.1011
[26]	Maxima 5.4, W. Schelter et al., http://maxima.sourceforge.net/
[27]	Polymake 2.0.1, E. Gavrilow, M. Joswig et al., http://www.math.tu-berlin.de/ polymake