Toric and tropical compactifications of hyperplane complements
Annales de la faculté des sciences de Toulouse Mathématiques (2014)
- Volume: 23, Issue: 2, page 297-333
- ISSN: 0240-2963
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topDenham, Graham. "Toric and tropical compactifications of hyperplane complements." Annales de la faculté des sciences de Toulouse Mathématiques 23.2 (2014): 297-333. <http://eudml.org/doc/275381>.
@article{Denham2014,
abstract = {These lecture notes survey and compare various compactifications of complex hyperplane arrangement complements. In particular, we review the Gel$^\{\prime \}$fand-MacPherson construction, Kapranov’s visible contours compactification, and De Concini and Procesi’s wonderful compactification. We explain how these constructions are unified by some ideas from the modern origins of tropical geometry.},
author = {Denham, Graham},
journal = {Annales de la faculté des sciences de Toulouse Mathématiques},
keywords = {hyperplane arrangement; compactification; matroid; toric variety; visible contours; wonderful compactification},
language = {eng},
month = {3},
number = {2},
pages = {297-333},
publisher = {Université Paul Sabatier, Toulouse},
title = {Toric and tropical compactifications of hyperplane complements},
url = {http://eudml.org/doc/275381},
volume = {23},
year = {2014},
}
TY - JOUR
AU - Denham, Graham
TI - Toric and tropical compactifications of hyperplane complements
JO - Annales de la faculté des sciences de Toulouse Mathématiques
DA - 2014/3//
PB - Université Paul Sabatier, Toulouse
VL - 23
IS - 2
SP - 297
EP - 333
AB - These lecture notes survey and compare various compactifications of complex hyperplane arrangement complements. In particular, we review the Gel$^{\prime }$fand-MacPherson construction, Kapranov’s visible contours compactification, and De Concini and Procesi’s wonderful compactification. We explain how these constructions are unified by some ideas from the modern origins of tropical geometry.
LA - eng
KW - hyperplane arrangement; compactification; matroid; toric variety; visible contours; wonderful compactification
UR - http://eudml.org/doc/275381
ER -
References
top- Ardila (F.), Benedetti (C.), Doker (J.).— Matroid polytopes and their volumes, Discrete Comput. Geom. 43, no. 4, p. 841-854 (2010). Zbl1204.52016MR2610473
- Ardila (F.), Klivans (C. J).— The Bergman complex of a matroid and phylogenetic trees, J. Combin. Theory Ser. B 96, no. 1, p. 38-49 (2006). Zbl1082.05021MR2185977
- Blasiak (J.).— The toric ideal of a graphic matroid is generated by quadrics, Combinatorica 28, no. 3, p. 283-297 (2008). Zbl1212.05030MR2433002
- Cohen (D. C.), Denham (G.), Falk (M. J.), Schenck (H. K.), Suciu (A. I.), Terao (H.), Yuzvinsky (S.).— Complex arrangements: algebra, geometry, topology, in preparation.
- Cohen (D. C.), Denham (G.), Falk (M. J.), Varchenko (A.).— Critical points and resonance of hyperplane arrangements, Canad. J. Math. 63, no. 5, p. 1038-1057 (2011). Zbl1228.32028MR2866070
- Catanese (F.), Hoşten (S.), Khetan (A.), Sturmfels (B.).— The maximum likelihood degree, Amer. J. Math. 128, no. 3, p. 671-697 (2006). Zbl1123.13019MR2230921
- Cox (D. A.), Little (J. B.), Schenck (H. K.).— Toric varieties, Graduate Studies in Mathematics, vol. 124, American Mathematical Society, Providence, RI (2011). Zbl1223.14001MR2810322
- De Concini (C.), Procesi (C.).— Wonderful models of subspace arrangements, Selecta Math. (N.S.) 1, no. 3, p. 459-494 (1995). Zbl0842.14038MR1366622
- Delucchi (E.), Dlugosch (M.).— Bergman complexes of lattice path matroids, arXiv:1207.4700. 5.2 Zbl1323.05028
- Denham (G.), Garrousian (M.), Schulze (M.).— A geometric deletion-restriction formula, Adv. Math. 230, no. 4-6, p. 1979-1994 (2012). Zbl1317.52032MR2927361
- Denham (G.), Garrousian (M.), Tohăneanu (Ş.).— Modular decomposition of the Orlik-Terao algebra of a hyperplane arrangement, Ann. Combin., to appear, arXiv:1211.4562. 5.1. Zbl1300.52019MR1771613
- Feichtner (E. M.).— De Concini-Procesi wonderful arrangement models: a discrete geometer’s point of view, Combinatorial and computational geometry, MSRI Publ., vol. 52, Cambridge Univ. Press, Cambridge, p. 333-360 (2005). Zbl1120.52009MR2178326
- Feichtner (E. M.), Kozlov (D. N.).— Incidence combinatorics of resolutions, Selecta Math. (N.S.) 10, no. 1, p. 37-60 (2004). Zbl1068.06004MR2061222
- Feichtner (E. M.), Müller (I.).— On the topology of nested set complexes, Proc. Amer. Math. Soc. 133, no. 4, p. 999-1006 (2005) (electronic). Zbl1053.05125MR2117200
- Feichtner (E. M.), Sturmfels (B.).— Matroid polytopes, nested sets and Bergman fans, Port. Math. (N.S.) 62, no. 4, p. 437-468 (2005). Zbl1092.52006MR2191630
- Feichtner (E. M.), Yuzvinsky (S.).— Chow rings of toric varieties defined by atomic lattices, Invent. Math. 155, no. 3, p. 515-536 (2004). Zbl1083.14059MR2038195
- Fulton (W.), MacPherson (R. D.).— A compactification of configuration spaces, Ann. of Math. (2) 139, no. 1, p. 183-225 (1994). Zbl0820.14037MR1259368
- Gelfand (I. M.), Goresky (M.), MacPherson (R. D.), Serganova (V. V.). — Combinatorial geometries, convex polyhedra, and Schubert cells, Adv. in Math. 63, no. 3, p. 301-316 (1987). Zbl0622.57014MR877789
- Gelfand (I. M.), Kapranov (M. M.), Zelevinsky (A. V.).— Discriminants, resultants, and multidimensional determinants, Mathematics: Theory & Applications, Birkhäuser Boston Inc., Boston, MA, (1994). Zbl1138.14001MR1264417
- Gelfand (I. M.), MacPherson (R. D.).— Geometry in Grassmannians and a generalization of the dilogarithm, Adv. in Math. 44, no. 3, p. 279-312 (1982). Zbl0504.57021MR658730
- Grayson (D.), Stillman (M.).— Macaulay2–a software system for algebraic geometry and commutative algebra, available at http://www.math.uiuc.edu/Macaulay2.
- Gaiffi (G.), Serventi (M.).— Families of building sets and regular wonderful models, European Jornal of Combinatorics, 36, p. 17-38 (2014). Zbl1326.14129MR3131872
- Hacking (P.).— The homology of tropical varieties, Collect. Math. 59, no. 3, p. 263-273 (2008). Zbl1198.14059MR2452307
- Horiuchi (H.), Terao (H.).— The Poincaré series of the algebra of rational functions which are regular outside hyperplanes, J. Algebra 266, no. 1, p. 169-179 (2003). Zbl1084.13501MR1994536
- Huh (J.), Katz (E.).— Log-concavity of characteristic polynomials and the Bergman fan of matroids, Math. Ann. 354, no. 3, p. 1103-1116 (2012). Zbl1258.05021MR2983081
- Huh (J.), Sturmfels (B.).— Likelihood Geometry, arXiv:1305.7462. 5.3. MR2524076
- Huh (J.).— The maximum likelihood degree of a very affine variety, Compos. Math., Compositio Math, 149, p. 1245-1266 (2013). Zbl1282.14007MR3103064
- Kapranov (M. M.).— Chow quotients of Grassmannians. I, I. M. Gelfand Seminar, Adv. Soviet Math., vol. 16, Amer. Math. Soc., Providence, RI, p. 29-110 (1993). Zbl0811.14043MR1237834
- Kashiwabara (K.).— The toric ideal of a matroid of rank 3 is generated by quadrics, Electron. J. Combin. 17, no. 1, Research Paper 28, 12 (2010). Zbl1205.05041MR2595488
- Katz (E.).— A tropical toolkit, Expo. Math. 27, no. 1, p. 1-36 (2009). Zbl1193.14004MR2503041
- Katz (E.).— Tropical intersection theory from toric varieties, Collect. Math. 63, no. 1, p. 29-44 (2012). Zbl1322.14091MR2887109
- Lenz (M.).— The f-vector of a realizable matroid complex is strictly log-concave, Advances in Applied Mathematics 51, no. 5, p. 543-454 (2013). Zbl1301.05382MR3118543
- Looijenga (E.).— Compactifications defined by arrangements. I. The ball quotient case, Duke Math. J. 118, no. 1, p. 151-187 (2003). Zbl1052.14036MR1978885
- Mikhalkin (G.).— Tropical geometry and its applications, ICM Lectures, Vol. II, Eur. Math. Soc., Zürich, p. 827-852 (2006). Zbl1103.14034MR2275625
- Orlik (P.), Terao (H.).— Arrangements of hyperplanes, Grundlehren der Mathematischen Wissenschaften, vol. 300, Springer-Verlag, Berlin (1992). Zbl0757.55001MR1217488
- Oxley (J.).— Matroid theory, second ed., Oxford Graduate Texts in Mathematics, vol. 21, Oxford University Press, Oxford (2011). Zbl1254.05002MR2849819
- Postnikov (A.).— Permutohedra, associahedra, and beyond, Int. Math. Res. Not. IMRN, no. 6, p. 1026-1106 (2009). Zbl1162.52007MR2487491
- Proudfoot (N. J.), Speyer (D.).— A broken circuit ring, Beiträge Algebra Geom. 47, no. 1, p. 161-166 (2006). Zbl1095.13024MR2246531
- Richter-Gebert (J.), Sturmfels (B.), Theobald (T.).— First steps in tropical geometry, Idempotent mathematics and mathematical physics, Contemp. Math., vol. 377, Amer. Math. Soc., Providence, RI, p. 289-317 (2005). Zbl1093.14080MR2149011
- Schrijver (A.).— Combinatorial optimization. Polyhedra and efficiency. Vol. B, Algorithms and Combinatorics, vol. 24, Springer-Verlag, Berlin, Matroids, trees, stable sets, Chapters p. 39-69 (2003). Zbl1041.90001MR1956925
- Sanyal (R.), Sturmfels (B.), Vinzant (C.).— The entropic discriminant, Advances in Mathematics 244, p. 678-707 (2013). MR3077886
- Schenck (H.), Tohăneanu (Ş O.).— The Orlik-Terao algebra and 2-formality, Math. Res. Lett. 16, no. 1, p. 171-182 (2009). Zbl1171.52011MR2480571
- Sturmfels (B.).— Gröbner bases and convex polytopes, University Lecture Series, vol. 8, American Mathematical Society, Providence, RI (1996). Zbl0856.13020MR1363949
- Sturmfels (B.).— Equations defining toric varieties, Algebraic geometry–Santa Cruz 1995, Proc. Sympos. Pure Math., vol. 62, Amer. Math. Soc., Providence, RI, p. 437-449 (1997). Zbl0914.14022MR1492542
- Sturmfels (B.).— Solving systems of polynomial equations, CBMS Regional Conference Series in Mathematics, vol. 97, Published for the Conference Board of the Mathematical Sciences, Washington, DC (2002). Zbl1101.13040MR1925796
- Suciu (A. I.).— Hyperplane arrangements and Milnor fibrations, this volume (2013).
- Tevelev (J.).— Compactifications of subvarieties of tori, Amer. J. Math. 129, no. 4, p. 1087-1104 (2007). Zbl1154.14039MR2343384
- Varchenko (A.).— Quantum integrable model of an arrangement of hyperplanes, SIGMA Symmetry Integrability Geom. Methods Appl. 7, Paper 032, 55 (2011). Zbl1217.82026MR2804564
- White (N. L.).— The basis monomial ring of a matroid, Advances in Math. 24, no. 3, p. 292-297 (1977). Zbl0357.05031MR437366
- White (N. L.).— A unique exchange property for bases, Linear Algebra Appl. 31, p. 81-91 (1980). Zbl0458.05022MR570381
- Ziegler (G. M.).— Lectures on polytopes, Graduate Texts in Mathematics, vol. 152, Springer-Verlag, New York (1995). Zbl0823.52002MR1311028
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