Informally, an algebraic variety is a geometric object. In algebraic geometry line bundles are particularly interesting as they can be. In algebraic geometry, a line bundle on a projective variety is nef if it has nonnegative degree on every curve in the variety. On triangular decompositions of algebraic varieties.
Algebraic geometry university of california, riverside. First, we obtain the good zariski decomposition via the anticanonical model. Diophantine approximation and a lifting theorem 9 4. 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. Classification of noncomplete algebraic varieties 417 424. 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. Demailly grenoble, tsimf, sanya, dec 1822, 2017 ricci curvature and geometry of compact kahler varieties 772. On zariski decomposition with and without support laface, roberto, taiwanese journal of mathematics, 2016. Some topics on zariski decompositions and restricted base loci of divisors on singular varieties tesi di dottorato in matematica di. Zariskis problem mathematisches institut universitat bonn. I am not familiar with examples of this technique in use though. Introduction to algebraic surfaces lecture notes for the course at the university of mainz wintersemester 20092010 arvid perego preliminary draft. Our goal is to understand several types of algebraic varieties. A more classical approach is to look for a zariski decomposition of d, i.
Moreno maza1 computational mathematics group, nag ltd, oxford ox2 8dr, greatbritain abstract di. 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. Zariski decomposition of divisors on algebraic varieties, duke math. Approximate analytic zariski decomposition and abundance. If zis any algebraic set, the zariski topology on zis the topology induced on it from an. Zariski decomposition of divisors on algebraic varieties. In this subsection, we study how the redundant mmp affects the geometry of the variety when the anticanonical divisor admits the good zariski decomposition. Zariski decompositions, volumes, and stable base loci uni frankfurt. More precisely, knowing the zariski decomposition of a qdivisor provides a quick. For computational methods on polynomials we refer to the books by. On triangular decompositions of algebraic varieties m. Knapp, advanced algebra, digital second edition east setauket, ny. Different variants of singular holomorphic symplectic varieties have been extensively studied in recent years.
The notion of zariski decomposition 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. Algebraic geometry, bowdoin, 1985 brunswick, maine, 1985. This is the second part of our work on zariski decomposition structures, where we compare two different volume type functions for curve classes. 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. Periods, moduli spaces and arithmetic of algebraic varieties, and the otka grant 61116 by the hungarian. For simplicity suppose that xis a nonsingular algebraic variety. In this note we first show that the boucksom zariski decomposition holds in the largest possible. Intersection multiplicities of holomorphic and algebraic curves with divisors noguchi, junjiro, 2004. Ample subvarieties of algebraic varieties, lecture notes in mathematics, vol. Yujiro kawamata the zariski decomposition of logcanonical divisors mr 927965 m. This generalizes the usual homogeneous coordinate ring of the projective space and constructions of cox and kajiwara for smooth and divisorial toric varieties. 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. In a 1962 paper, zariski introduced the decomposition theory that now bears his name.
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. In algebraic geometry, divisors are a generalization of codimension1 subvarieties of algebraic varieties. 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 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. Some topics on zariski decompositions and restricted base. Commutative algebra is the study of commutative rings and attendant structures, especially ideals and modules. We then treat the case of a surface and a hyperk\ahler manifold in some detail.
A usefull geometric tool is the group of divisors on a variety x. Proceedings of symposia in pure mathematics part 1, vol. N 2divs r is called azariski decomposition of d if the following conditions are satis ed. Factorization of anticanonical maps of fano type variety. Minkowskis existence theorem is the convex geometry version of the duality between the pseudoe ective cone of divisors and the movable cone of curves. At the same time it still remains unknown which nonnegative real algebraic numbers arise as volumes of cartier divisors on some variety.
Let s be a nonsingular projective surface over an algebraically closed eld. We explain how to use the covering trick to generalize the kodaira vanishing theorem for. 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. The volume of a cartier divisor d on a projective complex variety x measures the asymptotic rate of growth of. Pdf a simple proof for the existence of zariski decomposition on. 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. A ne nspace, an k, is a vector space of dimension n over k.
Positivity functions for curves on algebraic varieties. Pdf zariski decomposition of pseudoeffective divisors. The notion of zariskidecomposition introduced by oscar zariski is a powerful tool in the study of open surfaces. Especially projective algebraic varieties are kahler. Positivity functions for curves on algebraic varieties core. Zariski decomposition plays an important role in the theory of algebraic surfaces due to many applications. In this note we consider the problem of integrality of zariski decompositions for pseudoeffective integral divisors on algebraic surfaces. Using the intersection form respectively the beauvillebogomolov form, we characterize the modified nef cone and the exceptional divisors. 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. In this direction, one of the most basic results is. Boucksom showed that it also holds for irreducible symplectic manifolds. The zariski decomposition of logcanonical divisors.
Linear series on surfaces and zariski decomposition. Introduction the purpose of mori theory is to give a meaningful birational classi cation of a large class of algebraic varieties. One of fundamental problems of algebraic geometry is the question. By analogy with the algebraic morse inequality for nef divisors, we. In this note we first show that the boucksomzariski decomposition holds in the largest possible. Zariski decomposition of curves on algebraic varieties. Zariski decomposition in shokurovs sense and bssampleness 20. Subadditivity of multiplier ideals and fujitas approximate zariski decomposition153 chapter 15.
Cutkosky, zariski decomposition of divisors on algebraic varieties. Here we extend a construction first used by cutkosky, and use the theory of real multiplication on abelian varieties. Both are ultimately derived from the notion of divisibility in the integers and algebraic number fields. Closedopen sets in zare intersections of zwith closedopen sets in an. A divisor is an element of the free abelian group generated by the sub. On integral zariski decompositions of pseudoeffective. 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. Divisorial zariski decompositions on compact complex manifolds. 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. The zariski decomposition of logcanonical divisors 425 432.