Proceedings of symposia in pure mathematics part 1, vol. The aim of this section is to introduce the zariski decomposition, which will play an essential. Yujiro kawamata the zariski decomposition of logcanonical divisors mr 927965 m. Although it arose in the context of algebraic geometry and deals with the configuration of curves on an algebraic surface, we have recently observed that the essential. Zariski decomposition in shokurovs sense and bssampleness 20. In this direction, one of the most basic results is. Cutkosky, zariski decomposition of divisors on algebraic varieties. Moreno maza1 computational mathematics group, nag ltd, oxford ox2 8dr, greatbritain abstract di. For simplicity suppose that xis a nonsingular algebraic variety. Algebraic geometry, bowdoin, 1985 brunswick, maine, 1985. This is the only possible short answer i can think of, but it is not completely satisfying. On triangular decompositions of algebraic varieties m. At the same time it still remains unknown which nonnegative real algebraic numbers arise as volumes of cartier divisors on some variety.
Pdf a simple proof for the existence of zariski decomposition on. Periods, moduli spaces and arithmetic of algebraic varieties, and the otka grant. One central unifying concept is positivity, which can be viewed either in algebraic terms positivity of divisors and algebraic cycles, or in more analytic terms plurisubharmonicity, hermitian connections with positive curvature. Different variants of singular holomorphic symplectic varieties have been extensively studied in recent years. In this note we consider the problem of integrality of zariski decompositions for pseudoeffective integral divisors on algebraic surfaces. First, we obtain the good zariski decomposition via the anticanonical model. Knapp, advanced algebra, digital second edition east setauket, ny. Informally, an algebraic variety is a geometric object. Serre famously made use of the zariski topology to introduce sheaf cohomology to algebraic geometry, which was as i understand it a crucial innovation. The notion of zariskidecomposition introduced by oscar zariski is a powerful tool in the study of open surfaces.
To rst approximation, a projective variety is the locus of zeroes of a system of homogeneous polynomials. Linear series on surfaces and zariski decomposition. Moving to the zariski topology on schemes allows the use of generic points. Our goal is to understand several types of algebraic varieties. A ne nspace, an k, is a vector space of dimension n over k. In classical algebraic geometry that is, the part of algebraic geometry in which one does not use schemes, which were introduced by grothendieck around 1960, the zariski topology is defined on algebraic varieties. Demailly grenoble, tsimf, sanya, dec 1822, 2017 ricci curvature and geometry of compact kahler varieties 772. Algebraic geometry university of california, riverside. Both are ultimately derived from the notion of divisibility in the integers and algebraic number fields. If zis any algebraic set, the zariski topology on zis the topology induced on it from an. By analogy with the algebraic morse inequality for nef divisors, we. Introduction to algebraic surfaces lecture notes for the course at the university of mainz wintersemester 20092010 arvid perego preliminary draft.
A more classical approach is to look for a zariski decomposition of d, i. The first function is the polar transform of the volume for ample divisor classes. Boucksom showed that it also holds for irreducible symplectic manifolds. Here we extend a construction first used by cutkosky, and use the theory of real multiplication on abelian varieties. One of fundamental problems of algebraic geometry is the question. The classes of nef line bundles are described by a convex cone, and the possible contractions of the variety correspond to certain faces of the nef cone. In algebraic geometry, a line bundle on a projective variety is nef if it has nonnegative degree on every curve in the variety. A divisor is an element of the free abelian group generated by the sub.
In algebraic geometry, divisors are a generalization of codimension1 subvarieties of algebraic varieties. The divisorial zariski decomposition is orthogonal, and is thus a rational decomposition, which fact accounts for the usual existence statement of a zariski decomposition on a projective surface, which. Ample subvarieties of algebraic varieties, lecture notes in mathematics, vol. In this note we first show that the boucksomzariski decomposition holds in the largest possible. N 2divs r is called azariski decomposition of d if the following conditions are satis ed. The zariski decomposition of logcanonical divisors 425 432. Let s be a nonsingular projective surface over an algebraically closed eld. Introduction the purpose of mori theory is to give a meaningful birational classi cation of a large class of algebraic varieties. Periods, moduli spaces and arithmetic of algebraic varieties, and the otka grant 61116 by the hungarian. Zariski decomposition of divisors on algebraic varieties. For computational methods on polynomials we refer to the books by. I am not familiar with examples of this technique in use though.
Approximate analytic zariski decomposition and abundance. Commutative algebra is the study of commutative rings and attendant structures, especially ideals and modules. More precisely, knowing the zariski decomposition of a qdivisor provides a quick. Divisorial zariski decompositions on compact complex manifolds. Some topics on zariski decompositions and restricted base. In a 1962 paper, zariski introduced the decomposition theory that now bears his name. In this subsection, we study how the redundant mmp affects the geometry of the variety when the anticanonical divisor admits the good zariski decomposition. Pdf zariski decomposition of pseudoeffective divisors.
An analytic zariski decomposition of l is a singular metric h on l, semipositive in the sense of currents, such that for all k, h0. The zariski topology, defined on the points of the variety, is the topology such that the closed sets are the algebraic subsets of the variety. In algebraic geometry line bundles are particularly interesting as they can be. Positivity functions for curves on algebraic varieties core. When publishing the paper i was completely unaware that the calculation of the dimension of the chow variety of pn was done previously by pablo azcue in his 1992 thesis on the dimension of chow varieties under joe harris at harvard. Especially projective algebraic varieties are kahler. Linear series on surfaces and zariski decomposition this is an extended version of a talk given at the algebrageometry seminar at the university of freiburg in may 2011. We explain how to use the covering trick to generalize the kodaira vanishing theorem for. Zariski decomposition of curves on algebraic varieties. On integral zariski decompositions of pseudoeffective. The second function captures the asymptotic geometry of curves analogously to the volume function for divisors.
Asymptotic behavior of the dimension of the chow variety adv. Positivity functions for curves on algebraic varieties. Some topics on zariski decompositions and restricted base loci of divisors on singular varieties tesi di dottorato in matematica di. Minkowskis existence theorem is the convex geometry version of the duality between the pseudoe ective cone of divisors and the movable cone of curves. Zariski decomposition of bdivisors 3 shokurovs paper 18 and the survey article by prokurhov accompanying it 17 contain many interesting zariskitype decompositions, some of which work for b. Zariskis problem mathematisches institut universitat bonn. Diophantine approximation and a lifting theorem 9 4. Whether youve loved the book or not, if you give your honest and detailed thoughts then people will find new books that are right for them. Other readers will always be interested in your opinion of the books youve read. We might as well say that hamlet, prince of denmark is about a. This is the second part of our work on zariski decomposition structures, where we compare two different volume type functions for curve classes. Zariski decomposition of divisors on algebraic varieties, duke math. We then treat the case of a surface and a hyperk\ahler manifold in some detail. Zariski decomposition plays an important role in the theory of algebraic surfaces due to many applications.
Subadditivity of multiplier ideals and fujitas approximate zariski decomposition153 chapter 15. Classification of noncomplete algebraic varieties 417 424. Intersection multiplicities of holomorphic and algebraic curves with divisors noguchi, junjiro, 2004. This book is a collection of his works on the numerical aspects of divisors of algebraic varieties. In view of the correspondence between line bundles and divisors built from codimension1 subvarieties, there is. Factorization of anticanonical maps of fano type variety. The notion of zariski decomposition introduced by oscar zariski is a powerful tool in the study of open surfaces. On triangular decompositions of algebraic varieties. On zariski decomposition with and without support laface, roberto, taiwanese journal of mathematics, 2016. Zariski decompositions, volumes, and stable base loci uni frankfurt. Closedopen sets in zare intersections of zwith closedopen sets in an.
This generalizes the usual homogeneous coordinate ring of the projective space and constructions of cox and kajiwara for smooth and divisorial toric varieties. A usefull geometric tool is the group of divisors on a variety x. The volume of a cartier divisor d on a projective complex variety x measures the asymptotic rate of growth of. In this note we first show that the boucksom zariski decomposition holds in the largest possible. The zariski decomposition of logcanonical divisors. Hard lefschetz theorem with multiplier ideal sheaves. Using the intersection form respectively the beauvillebogomolov form, we characterize the modified nef cone and the exceptional divisors.1177 986 812 744 1149 1182 842 671 513 1528 31 30 1550 1053 1323 1348 696 1520 1422 229 789 1164 501 91 407 923 364 347 1509 387 1272 364 1248 1059 774 765 1193 663 10 684 1490 583 34 15 1158 631