本論文分三部分。第一部分我們證明了一個階化線性系統的體積與漸進自交數的關係。這個結果統一並推廣了一些過去文獻中出現的結果。第二個部分我們證明了數個與向量束的飯高維度有關的不變量在背景空間為投影曲線時是一樣的，並依此給出了[10]中一個問題的反例。第三部分我們討論如何計算環向量束的飯高維度。根據得到的結果，我們造出了兩個例子回答了[17]中的一個問題。

This thesis consists of three parts, which are under the same theme while being almost logically independent.
In the first part, we obtained a relation between the volume and the asymptotic self-intersection number of a graded linear series, unifying and generalizing several results in the literature.
In the second part, we proved that several invariants involved in the discussion of the Iitaka dimension of a vector bundle are the same, when the underlying space is a projective curve. Using the result, we constructed an example answering negatively a question of [10].
In the third part we discussed how to calculate the Iitaka dimension of a toric vector bundle. Base on the result, we constructed a couple of examples answering a question of [17].

Bibliography
[1] Michael F. Atiyah. On the krull-schmidt theorem with application to sheaves.
Bulletin de la Soci´ et´ e Math´ ematique de France, 84:307–317, 1956.
[2] Thomas Bauer, Sándor J. Kovács, Alex Küronya, Ernesto C. Mistretta, Tomasz
Szemberg, and Stefano Urbinati. On positivity and base loci of vector bundles.
European Journal of Mathematics, 1(2):229–249, 2015.
[3] L. D. Biagio and G. Pacienza. Restricted volumes of effective divisors. Bull. Soc.
Math. France, 144:299–337, 2016.
[4] Sebastien Boucksom, Jean-Pierre Demailly, Mihai Paun, and Thomas Peternell.
The pseudo-effective cone of a compact kähler manifold and varieties of negative
kodaira dimension. Journal of Algebraic Geometry, 22, 06 2004.
[5] David A. Cox, John B. Little, and Henry K. Schenck. Toric Varieties. American
Mathematical Soc., 2011.
[6] S. Di Rocco, K. Jabbusch, and G. G. Smith. Toric vector bundles and parliaments
of polytopes. Transactions of the American Mathematical Society, 370(11), 2014.
[7] Lawrence Ein, Robert Lazarsfeld, Mircea Mustat ¸˘ a, Michael Nakamaye, and Mi-
hnea Popa. Restricted volumes and base loci of linear series,. Amer. J. Math.,
131(6):607–651, 2009.
[8] M. Fulger. The cones of effective cycles on projective bundles over curves. Math-
ematische Zeitschrift, 264(3), 2009.
[9] Robin Hartshrone. Algebraic Geometry. Springer, New York, 1977.
[10] M. Hering, M. Mustat ¸ˇ a, and S. Payne. Positivity for toric vector bundles. Annales
de l’Institut Fourier, 60(2):607–640, 2010.
[11] Shin-Yow Jow. Okounkov bodies and restricted volumes along very general curves.
Adv. Math., 223(6):1356–1371, 2010.
54
BIBLIOGRAPHY 55
[12] Shin-Yow Jow. Iitaka dimensions of vector bundles. Annali di Matematica Pura
ed Applicata, 2018.
[13] k. Kaveh and A. Khovanskii. Newton-okounkov bodies, semigroups of integral
points, graded algebras and intersection theory. Ann. Math., 176:925–978, 2012.
[14] A. A. Klyachko. Equivariant vector bundles on toric varieties and some problems
of linear algebra. Banach Center Publications, 26(2):345–355, 1990.
[15] R.K. Lazarsfeld. Positivity in Algebraic Geometry I. Springer-Verlag Berlin Hei-
delberg, 1 edition, 2004.
[16] Robert Lazarsfeld and Mircea Mustat ¸˘ a. Convex bodies associated to linear series.
Ann. Sci. Ecole Norm. Sup., 42:783–835, 2009.
[17] E. C. Mistretta and S. Urbinati. Iitaka fibrations for vector bundles. IMRN,
rnx239, 2016.
[18] James G. Oxley. Matroid Theory. Oxford University Press, 1992.
[19] S. Payne. Moduli of toric vector bundles. Compositio Math. 1, 144:1199–1213,
2008.
[20] Sam Payne. Stable base loci, movable curves, and small modifications, for toric
varieties. Mathematische Zeitschrift, 253, 07 2005.