代數纖叢上的雙有理幾何(2/3)

行政院國家科學委員會專題研究計畫 期中進度報告

代數纖叢上的雙有理幾何(2/3)

中 華 民 國 94 年 5 月 18 日

一、中文摘要 我們證明對於一般形的三維極小模型, 若其奇點是高瑞斯坦且正典,則其上的第 五正典映射是雙有理。 關鍵詞：線性系、重正典映射、傅利葉-向井變換、 非正規流形。 Abstract

Let X be a complex projective minimal Gorenstein 3-fold of general type with canonical singularities. We prove that the 5-canonical map is birational onto its image.. Keywords: linear series, pluricanonical map

Fourier-Mukai transform, irregular varieties.

二、緣由與目的

One main goal of algebraic geometry is to classify algebraic varieties. The successful 3-dimensional MMP has been attracting more and more mathematicians to the study of algebraic 3-folds. In this paper, we restrict our interest to projective minimal Gorenstein 3-folds X of general type where there still remain many open problems.

Denote by KX the canonical divisor and

Φm := ΦmKX the m-canonical map. There

have been a lot of works along the line of the canonical classification. For instance, when

X is a smooth 3-fold of general type with the

pluri-genus h0(X, k KX) ≥ 2, in [Ko], as an

application to his research on higher direct images of dualizing sheaves, Kollár proved thatΦm, with m=11k+5, is birational onto its

image. This result was improved by Meng Chen [Ch] to include the cases m with m≥

5k+6.

On the other hand, for 3-folds X of general type with q(X)>0, Kollár [Ko] first proved that Φ225 is birational.

Recently, a joint work with Hacon [CH] proved thatΦm is birational for m ≥ 7 by using

the Fourier-Mukai transform. Moreover, Luo [Lu] has some results for 3-folds of general type with h2( OX)>0.

Now for minimal and smooth projective 3-folds, it has been established that Φm, m

≥6 is a birational morphism onto its image

after 20 year long research by Wilson in the year 1980, Benveniste [Be] in the year 1986 (m≥ 8 ), Matsuki [Ma] in the year 1986 (m=7), Meng Chen [Ch] in the year 1998 (m= 6) and independently by Lee [Le], in the years 1999 (m=6; and also the base point freeness of m-canonical system for m≥4). A

very natural and well-known question arises: Question: Let X be a minimal Gorenstein 3-fold of general type. Is Φ5 birational onto

its images?

One reason to account for this is that the non-birationality of the 4-canonical system for surfaces may happen when they have smaller pg or K2 (see Bombieri [Bo]), whence

a naïve induction on the dimension would predict the non-birationality of the 5-canonical system on certain 3-folds with smaller invariants.

Nevertheless, there are also evidences supporting the birationality of Φ5 for

Gorenstein minimal 3-folds X of general type. For instance, one sees that K3 ≥2 for minimal and smooth X. So an analogy of Fujita's conjecture would predict that |5KX | gives a

birational map. We recall that Fujita's conjecture (the freeness part) has been proved by Fujita, Ein-Lazarsfeld and Kawamta when dim X≤ 4.

三、結論與討論

Our main result is the following:

Theorem. Let X be a projective minimal Gorenstein 3-fold of general type with canonical singularities. Then the m-canonical map Φm is a birational morphism onto its

image for all m≥5.

Example. The numerical bound "5" in Theorem is optimal.

There are plenty of supporting examples. For instance, let f:VÆB be any fibration where V is a smooth projective 3-fold of general type and B a smooth curve.

Assume that a general fiber of f has the minimal model S with KS2=1 and pg(S)=2.

(For example, take the product.) Then Φ4 is

apparently not birational (see [Bo]).

The main technique involve is a partial resolution, together with the Kawamata Viehweg vanishing theorem. We remark that 1

by using the similar trick in [CH], one can prove a slightly weaker result with simpler argument for irregular threefolds.

四、參考文獻

[Be] Benveniste, X., Sur les applications

pluricanoniques des varietes de type tres general en dimension 3. Math.

Ann. 320 (2001), 367-380.

[Bo] Bombieri, E., Canonical models of

surfaces of general type. Inst. Hautes

Eudes Sci. Publ. Math. 42 (1973), 171-219.

[Ch] Chen, M.. On pluricanonical maps

forthreefolds of general type, J. Math.

Soc. Japan 50 (1998), 615-621.

[CH] Chen, Jungkai A.; Hacon, Christopher, Linear series of

irregular varieties. Algebraic

Geometry in East Asia, Japan, 2002, World Scientific Press

[Ko] J. Kollár, Higher direct images of

dualizing sheaves I, Ann. Math. 123

(1986), 11-42; II, ibid. 124 (1986), 171-202.

[Le] Lee, Remarks on the pluricanonical

and adjoint linear series on projective threefolds, Commun.

Algebra 27 (1999), 4459-4476. [Lu] T. Luo, Global $2$-forms on regular

3-folds of general type, Duke Math. J. 71 (1993), no. 3, 859-869.

[Ma] K. Matsuki, On pluricanonical maps for 3-folds of general type, J. Math. Soc. Japan 38 (1986), 339-359.