代數纖維空間中正則除子與反正則除子的正性表現

Abstract

本論文的主要目的是探討在代數纖維空間中正則除子與反正則除子的正性表現，首先我們會證明給定一個代數纖維空間f :X→Y,若對X上的奇異點有一些限制，X的反正則除子−KX為一擬有效/Q-有效/巨大除子，且−KX之相對應的的漸進基底點集合並不支配Y 的話，則−KY 也會是擬有效/Q-有效/巨大除子。作為本結果的推論，我們可以證明若X僅具備川又對數端末(klt)奇異點/對數正規(lc) 奇異點，則f 的相對反正則除子−KX/Y 必然不是豐富除子/巨大nef除子。

第二部分我們會先證明對於滿足特定條件的代數纖維空間，其介於X以及F 的反正則切斷環之間的限制映射會是一個賦次環之間的環單射。該定理為江尻-權業單射性定理的一個變形。做為此一單射性定理的應用,我們會證明一個反正則除子版本的飯高猜想，其具體敘述如下：給定一代數纖維空間f :X →Y, 若X僅具有klt Q-Gorenstein奇異點,−KX 為一Q-有效除子,且其穩定基底點集合 B(−KX) 並不支配Y 的話，則對於f 的一般纖維F,我們有以下的不等式 κ(−KX) ≤ κ(−KF)+κ(−KY)。

於第三部分我們則會證明一般化非消滅猜想的部分特例並討論其與飯高猜想的關聯。一般化非消滅猜想敘述為給定一klt對(X,∆),X上的一個nefQ-Cartier除子L,以及一非負的有理數t,若KX+∆+tL為一擬有效Q-因子,則其會與某一有效Q-因子數值同值。於本論文中我們會證明對於有非負小平次元的三維多樣體, 若其小平次元與q(X)兩不變量當中有至少一個大於零,則該多樣體滿足一般化非消滅猜想。做為此結論的應用,我們可以證明對於代數纖維空間f :X→Y,若X的維度不超過七維且已知X有非負的小平維度,則該代數纖維空間滿足飯高猜想,也即是滿足以下不等式 κ(X) ≥ κ(F)+κ(Y)。

最後,我們會再額外補充一些有關反正則除子具有良好正性的代數多樣體的一些相關討論。

In this thesis, we will discuss the positivity behavior of the canonical divisor and the anticanonical divisor in algebraic fibre spaces. At first, we will prove for an algebraic fibre space f : X → Y, if X has mild singularities, −KX is pseudoeffective (resp. effective, big), and the corresponding asymptotic base locus of −KX does not dominate Y , then −KY isalso pseudoeffective (resp. effective, big). As a corollary, we can prove if X has klt (resp. lc) singularities, then the relative anticanonical divisor −KX/Y could not be nef and big (resp. could not be ample).

Secondly, we will prove for certainly algebraic fibre spaces, the restriction map between the anticanonical section ring of X and F is an injective graded ring homomorphism. This theorem is a variant of Ejiri-Gongyo’s Injectivity Theorem. As an application, we will prove an ”anticanonical version” of the Iitaka Conjecture. The statement is:if X has at worst klt Q-Gorenstein singularities, and −KX is Q-effective with the stable base locus B(−KX) does not dominate Y, the for a general fibre F of f, we have the inequality κ(−KX) ≤ κ(−KF)+κ(−KY).

Thirdly, we will prove some special cases of the Generalized Nonvanishing Conjecture and discuss its relation with the Iitaka Conjecture. The Generalized Nonvanishing Conjecture states that for a klt pair (X,∆), a nef Q-Cartier divisor L on X, and a non-negative rational number t, if KX +∆+tL is pseudoeffective, then it should numerically equivalent to an effective Q-divisor. In this thesis, we will prove that for threefolds with non-negative Kodaira dimension, if either the Kodaira dimension or the irregularity is positive, then the Generalized Nonvanishing Conjecture holds. As an application, we can show the Iitaka conjecture is true if X has non-negative Kodaira dimension and the dimension of X is at most seven, that is, for an algebraic fibre space f : X → Y with general fibre F, if dimX ≤ 7 and κ(X) ≥ 0, then we have κ(X) ≥ κ(F)+κ(Y).

Finally, we will give some other miscellanies results about varieties whose anticanonical divisor has good positivities.