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本期开始加开窗口,推出科学网特色博主,有用链接等。
他将自己的创造力归因于...一个孩子的天真而热情的好奇心*。
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(接上回*)Proof. (of Corollary 1.2) Since Δ is big, Δ ~R A + D where A is ample and D ≥ 0. Thus there is α ∈ (0, 1) such that if we let
Γ =(1 - α) Δ + α D,
then (X, Γ) is ε/2 - lc and -(Kx + Δ) is ample. Now apply Theorem 1.1.
评论:证明含有4个推导,看上去并不显然。(换句话说:看不懂)。
跳点:
1. 推论1.2的表述; (做卡片)
2. “big” 的含义;
3. “ample”的含义;
温习:
推论1.2的表述:{Xp} 有界,若 (X, Δ) 是 “eps-lc 零扩副大边”。
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小结:看上去该了解点基本概念了...现在就开始!(见如下外网摘录)
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A line bundle on a projective variety is ample if some tensor power of it is very ample.(链接).
An ample line bundle is one such that some positive power is very ample.(链接).
A line bundle L on a variety X is very ample if X can be embedded into a projective space so that L is the restriction of Serre's twisting sheaf O(1) on the projective space.
a very ample line bundle is one with enough global sections to set up an embedding of its base variety or manifold into projective space.
评论:两个基本概念“ample”和“very ample”,用于描述“line bundle”的性质。看上去“very ample”更为基本。
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A big line bundle L on X of dimension n is a line bundle such that .(链接)
A line bundle is big if it is of maximal Iitaka dimension, that is, if its Iitaka dimension is equal to the dimension of the underlying variety. Bigness is a birational invariant: If f : Y → X is a birational morphism of varieties, and if L is a big line bundle on X, then f*L is a big line bundle on Y. (链接)
All ample line bundles are big.
Big line bundles need not determine birational isomorphisms of X with its image. For example, if C is a hyperelliptic curve (such as a curve of genus two), then its canonical bundle is big, but the rational map it determines is not a birational isomorphism. Instead, it is a two-to-one cover of the canonical curve of C, which is a rational normal curve.
评论:“big”也是用来描述“line bundle”的性质。
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a line bundle on a complete algebraic variety over a field is said to be nef if the degree of its restriction to every algebraic curve in the variety is non-negative. The term "nef" was introduced by Miles Reid[1] as a replacement for the older terms "arithmetically effective" (Zariski 1962, definition 7.6) and "numerically effective", as well as for the phrase "numerically eventually free". (A line bundle is called semi-ample or "eventually free" if some positive power is basepoint-free.) (链接)
Every semi-ample divisor is nef, but not every nef divisor is numerically equivalent to a semi-ample divisor, or even to an effective divisor.
评论:“nef”仍然是“line bundle”的性质。
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