The Generic Finiteness Of The M-canonical Map For 3-folds Of General Type

To classify algebraic varieties is one of the goals in algebraicgeometry. One way to study a given variety is to understand thebehavior of its pluricanonical maps. The so-called m-canonicalmapφ_m is nothing but the rational map corresponding to thecomplete linear system |mK_X|. During the past years, the theoryof the smooth varieties over complex field with dim≤2 has beenimproved a large step by a long list of authors. As for the higherdimensional varieties, unfortunately, the classification is still atoo diffcult problem, although it has been developed a great dealby T. Fujita, K. Ueno, E. Viehweg, Y. Kawamata, J. Kollár, M. Reid,S. Mori and others. For a given 3-fold X, quite an interesting thingis to find the optimal m, such thatφ_k is birational or genericallyfinite(where k≥m) and we also hope such m is independent withX. We say thatφ_m is canonically stable. To find above m, weusually give more precise condition to canonical index, geometricgenus, canonical divisor etc. Little by little, we have some results.This thesis focuses on the generic finiteness of the m-canonicalmap for 3-folds of general type. In 1986, J. Kollár(Theorem(6.2)of [14]) gave an effective result and proved that it is optimal underthat condition. And in 2001, Meng Chen proved that for a projec-tive minimal Gorenstein threefold of general type,φ₃ is genericallyfinite whenever P_g(X)≥39. We generalize the method used in[5] and then improve the results of J. Kollár and Meng Chen.The thesis carries out the discussion in three chapters. In chapter one, we first introduce the two theorem mentionedabove and our main theorem. Then we present several well-knownresults which will be used throughout this paper. Finally, we willfix our terminology.In chapter two, we prove the main theorem step by step.In chapter three, we mainly dicuss the generical finitenessofφ_m through the geometric genus of 3-fold X and give moredetailed results.