C. Araujo and S. Druel, On codimension 1 del Pezzo foliations on varieties with mild singularities, And85] T. Ando. On extremal rays of the higher dimensional varieties. Invent. Math, pp.3-4769, 1985.
DOI : 10.1007/BF01389326

URL : https://hal.archives-ouvertes.fr/hal-01330035

R. [. Alexeev and . Pardini, On the existence of ramified Abelian covers, Rend. Semin. Mat, vol.71, pp.3-4307, 2013.

J. [. Araujo and . Ramón-marí, Flat deformations of P n, Bull. Braz. Math. Soc. (N.S.), issue.3, pp.45371-383, 2014.

J. [. Andreatta and . Wi´sniewskiwi´sniewski, A note on nonvanishing and applications. Duke Math, J, vol.72, issue.3, pp.739-755, 1993.

J. [. Andreatta and . Wi´sniewskiwi´sniewski, A view on contractions of higher-dimensional varieties, Algebraic geometry. Proceedings of the Summer Research Institute, pp.153-183, 1995.
DOI : 10.1090/pspum/062.1/1492522

]. V. Bat94 and . Batyrev, Dual polyhedra and mirror symmetry for Calabi- Yau hypersurfaces in toric varieties, J. Algebr. Geom, vol.3, issue.3, pp.493-535, 1994.

P. [. Birkar, C. D. Cascini, J. Hacon, and . Mckernan, Existence of minimal models for varieties of log general type, Journal of the American Mathematical Society, vol.23, issue.2, pp.405-468, 2010.
DOI : 10.1090/S0894-0347-09-00649-3

F. [. Bonavero, J. A. Campana, and . Wi´sniewskiwi´sniewski, Vari??t??s complexes dont l'??clat??e en un point est de Fano, Comptes Rendus Mathematique, vol.334, issue.6, pp.463-468, 2002.
DOI : 10.1016/S1631-073X(02)02284-7

]. C. Bir16 and . Birkar, Singularities of linear systems and boundedness of Fano varieties. Electronic preprint, 2016.

]. L. Bon02 and . Bonavero, Toric varieties whose blow-up at a point is Fano, Tohoku Math. J, vol.54, issue.24, pp.593-597, 2002.

]. S. Bou04 and . Boucksom, Divisorial Zariski decompositions on compact complex manifolds, Bou12] S. Boucksom. Corps d'Okounkov. Séminaire Bourbaki, pp.45-761, 2004.

]. M. Bri05 and . Brion, Lectures on the Geometry of Flag Varieties, Trends in Mathematics Basel: Birkhäuser, 2005.

A. [. Beltrametti and . Sommese, The adjunction theory of complex projective varieties, 1995.
DOI : 10.1515/9783110871746

]. W. Buc08 and . Buczynska, Fake weighted projective spaces. Electronic preprint, 2008.

]. D. But94 and . Butler, Normal generation of vector bundles over a curve

]. C. Cas09 and . Casagrande, On Fano manifolds with a birational contraction sending a divisor to a curve, Cat07] F. Catanese. Q.E.D. for algebraic varieties, pp.783-80543, 2007.

]. C. Cd08, S. Casagrande, and . Di-rocco, Projective Q-factorial toric varieties covered by lines, Commun. Contemp. Math, vol.10, issue.3, pp.363-389, 2008.

S. [. Casagrande and . Druel, Locally Unsplit Families of Rational Curves of Large Anticanonical Degree on Fano Manifolds, International Mathematics Research Notices, vol.2015, issue.21, pp.201510756-10800, 2015.
DOI : 10.1093/imrn/rnv011

URL : https://hal.archives-ouvertes.fr/hal-01330034

]. H. Che11 and . Chen, Computing volume function on projective bundle over a curve. Hodge theory and algebraic geometry, RIMS Kôkyûroku, vol.1745, pp.169-182, 2011.

S. [. Cox and . Katz, Mirror symmetry and algebraic geometry, 1999.
DOI : 10.1090/surv/068

J. [. Cox, H. K. Little, and . Schenck, Toric varieties, 2011.
DOI : 10.1090/gsm/124

]. S. Cut88 and . Cutkosky, Elementary contractions of Gorenstein threefolds, Math. Ann, vol.280, issue.3, pp.521-525, 1988.

]. D. Dai02 and . Dais, Resolving 3-dimensional toric singularities In Geometry of toric varieties. Lectures of the summer school, pp.155-186, 2000.

]. G. Del14 and . Della-noce, On the Picard number of singular Fano varieties, Int. Math. Res. Not, vol.2014, issue.4, pp.955-990, 2014.

R. L. Ein, M. Lazarsfeld, M. Musta¸t?musta¸t?-a, M. Nakamaye, and . Popa, Invariants asymptotiques des lieux de base, Annales de l???institut Fourier, vol.56, issue.6, pp.1701-1734, 2006.
DOI : 10.5802/aif.2225

R. L. Ein, M. Lazarsfeld, M. Musta¸t?musta¸t?-a, M. Nakamaye, and . Popa, Restricted volumes and base loci of linear series, Am. J. Math, vol.131, issue.3, pp.607-651, 2009.

. Fl-]-m, B. Fulger, and . Lehmann, Zariski decompositions of numerical cycle classes, J. Algebraic Geometry

]. O. Fuj99 and . Fujino, Applications of Kawamata's positivity theorem, Proc

]. O. Fuj06 and . Fujino, Equivariant completions of toric contraction morphisms, Tohoku Math. J, vol.58, issue.23, pp.303-321, 2006.

]. W. Ful84 and . Fulton, Intersection theory, Ergebnisse der Mathematik und ihrer Grenzgebiete, 3. Folge, Bd. 2. Berlin etc

]. M. Ful11 and . Fulger, The cones of effective cycles on projective bundles over curves, Math. Z, vol.269, issue.12, pp.449-459, 2011.

S. [. Greb, T. Kebekus, and . Peternell, ??tale fundamental groups of Kawamata log terminal spaces, flat sheaves, and quotients of abelian varieties, Duke Mathematical Journal, vol.165, issue.10, pp.1965-2004, 2016.
DOI : 10.1215/00127094-3450859

. A. Egaii and . Grothendieck, ´ Eléments de géométrie algébrique. I: Le langage des schémas. II: ´ Etude globalé elémentaire de quelques classe de morphismes. III: ´ Etude cohomologique des faisceaux cohérents premì ere partie), Publ. Math., Inst. HautesÉtudHautes´HautesÉtud. Sci, vol.4, pp.1-228, 1960.

]. D. Macaulay2, M. E. Grayson, and . Stillman, Macaulay2, a software system for research in algebraic geometry

]. R. Har71 and . Hartshorne, Ample vector bundles on curves, Nagoya Math. J, vol.43, pp.73-89, 1971.

S. [. Hu and . Keel, Mori dream spaces and GIT., The Michigan Mathematical Journal, vol.48, issue.1, pp.331-348, 2000.
DOI : 10.1307/mmj/1030132722

]. S. Ish91 and . Ishii, Quasi-Gorenstein Fano 3-folds with isolated non-rational loci, Compos. Math, vol.77, issue.3, pp.335-341, 1991.

]. V. Isk77 and . Iskovskikh, Fano 3-folds, I. Math. USSR, Izv, vol.11, pp.485-527, 1977.

]. V. Isk78 and . Iskovskikh, Fano 3-folds, II. Math. USSR, Izv, vol.12, pp.469-506, 1978.

]. V. Isk80 and . Iskovskikh, Anticanonical models of three-dimensional algebraic varieties, J. Sov. Math, vol.13, pp.745-814, 1980.

[. Jow, Okounkov bodies and restricted volumes along very general curves, Advances in Mathematics, vol.223, issue.4, pp.1356-1371, 2010.
DOI : 10.1016/j.aim.2009.09.015

]. K. Kar99 and . Karu, Semistable reduction in characteristic zero, 1999.

]. A. Kas13 and . Kasprzyk, Classifying terminal weighted projective space. Electronic preprint arXiv:1304, 2013.

]. K. Kat89 and . Kato, Logarithmic structures of Fontaine-Illusie In Algebraic analysis, geometry, and number theory: proceedings of the JAMI inaugural conference, pp.191-224, 1988.

]. Y. Kaw88 and . Kawamata, Crepant blowing-up of 3-dimensional canonical singularities and its application to degenerations of surfaces, Ann. Math, vol.127, issue.21, pp.93-163, 1988.

]. Y. Kaw06 and . Kawamata, Derived categories of toric varieties, Mich. Math. J, vol.54, issue.3, pp.517-535, 2006.

A. [. Kaveh and . Khovanskii, Newton-Okounkov bodies, semigroups of integral points, graded algebras and intersection theory, Annals of Mathematics, vol.176, issue.2
DOI : 10.4007/annals.2012.176.2.5

V. [. Küronya and . Lozovanu, Positivity of line bundles and Newton-Okounkov bodies. Electronic preprint arXiv, 2015.

]. S. Kle66 and . Kleiman, Toward a numerical theory of ampleness, Ann. Math, vol.84, issue.2, pp.293-344, 1966.

V. [. Küronya, C. Lozovanu, and . Maclean, Convex bodies appearing as Okounkov bodies of divisors, Advances in Mathematics, vol.229, issue.5, pp.2622-2639, 2012.
DOI : 10.1016/j.aim.2012.01.013

S. [. Kollár and . Mori, Birational geometry of algebraic varieties. With the collaboration of C. H. Clemens and A. Corti, 1998.

K. [. Kawamata, K. Matsuda, and . Matsuki, Introduction to the minimal model problem. Algebraic geometry, Proc. Symp., Sendai/Jap, pp.283-360, 1985.

Y. [. Kollár, S. Miyaoka, and . Mori, Rational connectedness and boundedness of Fano manifolds, Journal of Differential Geometry, vol.36, issue.3, pp.765-779, 1992.
DOI : 10.4310/jdg/1214453188

]. J. Kol96 and . Kollár, Rational curves on algebraic varieties, 1996.

]. J. Kol97 and . Kollár, Singularities of pairs, Algebraic geometry. Proceedings of the Summer Research Institute, pp.221-287, 1995.

]. J. Kol13 and . Kollár, Singularities of the Minimal Model Program. With the collaboration of Sándor Kovács, 2013.

]. R. Laz04 and . Lazarsfeld, Positivity in Algebraic Geometry, I & II, 2004.

R. Lazarsfeld and M. Musta¸t?musta¸t?, Convex bodies associated to linear series, Annales scientifiques de l'??cole normale sup??rieure, vol.42, issue.5, pp.783-835, 2009.
DOI : 10.24033/asens.2109

[. Potier, Lectures on vector bundles, 1997.

]. L. Man98 and . Manivel, Fonctions symétriques, polynômes de Schubert et lieux de dégénérescence, 1998.

]. K. Mat02 and . Matsuki, Introduction to the Mori program, 2002.

F. [. Muñoz, L. E. Sciullo, and . Solá-conde, On the existence of a weak Zariski decomposition on projectivized vector bundles, Geometriae Dedicata, vol.2, issue.76, pp.287-301, 2015.
DOI : 10.2307/1970376

]. R. Mir85 and . Miranda, Gorenstein toric threefolds with isolated singularities and cyclic divisor class group The Chern classes and Kodaira dimension of a minimal variety. Algebraic geometry, Proc. Symp., Sendai/Jap, pp.39-45, 1985.

S. [. Mori and . Mukai, Classification of Fano 3-folds with B 2 ?2, manuscripta mathematica, vol.110, issue.3, pp.147-162, 1981.
DOI : 10.1007/s00229-002-0336-2

S. [. Mori, . Mukai, and . Erratum, Classification of Fano 3-folds with B 2 ?2, manuscripta mathematica, vol.110, issue.3, pp.147-162407, 1981.
DOI : 10.1007/s00229-002-0336-2

]. S. Mol16 and . Molcho, Universal weak semistable reduction. Electronic preprint, 2016.

]. P. Mon16 and . Montero, On singular Fano varieties with a divisor of Picard number one Electronic preprint, p.2016

]. P. Mon17 and . Montero, Newton-Okounkov bodies on projective bundles over curves. Electronic preprint, 2017.

]. S. Mor75 and . Mori, On a generalization of complete intersections, J. Math

]. B. Nil05 and . Nill, Gorenstein toric Fano varieties, Manuscr. Math, vol.116, issue.2, pp.183-210, 2005.

]. A. Oko96 and . Okounkov, Brunn-Minkowski inequality for multiplicities, Invent . Math, vol.125, issue.3, pp.405-411, 1996.

]. A. Oko03 and . Okounkov, Why would multiplicities be log-concave? In The orbit method in geometry and physics, honor of A. A. Kirillov. Papers from the international conference, pp.329-347, 2000.

. G. Yu, V. V. Prokhorov, and . Shokurov, Towards the second main theorem on complements, J. Algebr. Geom, vol.18, issue.1, pp.151-199, 2009.

. Sgai-]-m and . Raynaud, Séminaire de géométrie algébrique du Bois MarieSGA 1), dirigé par Alexander Grothendieck Augmenté de deux exposés de M. Raynaud. RevêtementsRevêtementsétales et groupe fondamental. Exposés I ` a XIII. (Seminar on algebraic geometry at Bois Marie, directed by Alexander Grothendieck. Enlarged by two reports of M. Raynaud. ` Etale coverings and fundamental group). Lecture Notes in Mathematics, p.61, 1960.

]. M. Rei87 and . Reid, Young person's guide to canonical singularities Algebraic geometry, Proc. Symp. Pure Math, pp.345-414, 1985.

A. [. Ramanan and . Ramanathan, Some remarks on the instability flag, Tohoku Mathematical Journal, vol.36, issue.2, pp.269-291, 1984.
DOI : 10.2748/tmj/1178228852

V. [. Ravindra and . Srinivas, The Grothendieck-Lefschetz theorem for normal projective varieties, Journal of Algebraic Geometry, vol.15, issue.3, pp.563-590, 2006.
DOI : 10.1090/S1056-3911-05-00421-2

L. [. Rossi and . Terracini, -linear Gale duality and poly weighted spaces (PWS), Linear Algebra and its Applications, vol.495, pp.256-288, 2016.
DOI : 10.1016/j.laa.2016.01.039

URL : https://hal.archives-ouvertes.fr/hal-00304475

L. [. Rossi and . Terracini, A $$\mathbb {Q}$$ Q -factorial complete toric variety is a quotient of a poly weighted space, Annali di Matematica Pura ed Applicata (1923 -), vol.32, issue.1, pp.325-347, 2017.
DOI : 10.1016/j.laa.2016.01.039

]. T. Tsu06 and . Tsukioka, Classification of Fano manifolds containing a negative divisor isomorphic to projective space, Geom. Dedicata, vol.123, pp.179-186, 2006.

]. J. Wis91 and . Wisniewski, On contractions of extremal rays of Fano manifolds, J. Reine Angew. Math, vol.417, pp.141-157, 1991.

]. A. Wol05 and . Wolfe, Asymptotic invariants of graded systems of ideals and linear systems on projective bundles, 2005.