Pull back of cartier divisor
WebA relative effective Cartier divisor is an effective Cartier divisor D ˆX such that the projection D !X is flat. We will show that this notion is well behaved under base-change by any S0!S. … WebIn this case the linear system can be recovered by pulling back the hyperplane sections of Y ˆPm 1 and in fact O X(D) = ˚O Pn(1). 1. De nition 3.2. ... Let Dbe a Q-Cartier divisor on Y. (1) If Dis ample and fis nite then f Dis ample. (2) If f is surjective and f Dis ample (this can only happen if f is nite) then Dis ample.
Pull back of cartier divisor
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WebCwith a Cartier divisor. The clumsy way to do this is to proceed as above, and deform the divisor to a linearly equivalent divisor, which does not contain the curve. A more sophisticated approach is as follows. If the image of the curve lies in the divisor, then instead of pulling the divisor back, pullback the associated line bundle and take ... Webof ideals on Z (i.e. the pullback of O/I is an effective Cartier divisor), then there exists a unique morphism g : Z → X˜ factoring f. Z _ g_ _// f >˚˚ >>> >>> > X˜ π X In other words, if you have a morphism to X, which, when you pull back the ideal I, you get an effective Cartier divisor, then this factors through X˜ → X.
WebDefinition 31.26.2. Let X be a locally Noetherian integral scheme. A prime divisor is an integral closed subscheme Z \subset X of codimension 1. A Weil divisor is a formal sum D = \sum n_ Z Z where the sum is over prime divisors of X and the collection \ { Z \mid n_ Z \not= 0\} is locally finite (Topology, Definition 5.28.4 ). WebA relative effective Cartier divisor is an effective Cartier divisor D ˆX such that the projection D !X is flat. We will show that this notion is well behaved under base-change by any S0!S. Lemma 1. Suppose D ˆX is a relative effective Cartier divisor for f : X !S. For any S0!S, denote by f0: X0!S0the pullback. Then D0= S0 S D ˆX0is a ...
Web1.4. For a rational 1-contraction α: X99K Y, we may define the pull-back of any R-Cartier divisor Das follows: α∗D def= g ∗h ∗D(it is easy to show that this definition does not depend on the choice of the hut (1.2)). Note however that the map α∗ is not functorial: it is possible that (α β)∗ does not coincide with β∗α∗. Web31.13. Effective Cartier divisors. We define the notion of an effective Cartier divisor before any other type of divisor. Definition 31.13.1. Let be a scheme. A locally principal closed …
WebLemma : Let f: Y → X be a proper morphism of varieties such that that. R f ∗ O X = O Y. Let E be a Cartier divisor on Y. Then E is the pull back of a Cartier divisor on X if and only if for all x ∈ X, there is a neighborhood U of x in X such that E restricted to f − 1 ( U) is trivial. Let x ∈ X, and let U be a contractible ...
theater bartholomäbergWebLet B Z X denote the blow-up of X along Z and E Z ⊂ B Z X the exceptional divisor. We refer to π: B Z X → X as a blow-up if we imagine that B Z X is created from X, and a blow-down if we start with B Z X and construct X later. Note that E Z has codimension 1 and Z has codimension ≥ 2. Thus a blow-down decreases the Picard number by 1. theater bar stoolshttp://math.stanford.edu/~vakil/245/245class6.pdf the godfather part ii screenplay pdfWebDec 1, 2015 · Suppose that f: X → Z is a surjective morphism of normal varieties with connected fibers. Then an R -Cartier divisor L on X is f -numerically trivial if and only if there is an R -Weil divisor D on Z such that D is numerically Q -Cartier and f ⊛ D ≡ L where f ⊛ is the numerical pullback of [14]. The proof runs as follows. the godfather part iii soundtrackWebThe group of Cartier divisors on Xis denoted Div(X). 2.5. Some notation. To more closely echo the notation for Weil divisors, we will often denote a Cartier divisor by a single … the godfather part iii runtimeWebApr 6, 2024 · If there is a nontrivial linear relation among the Cartier divisor classes $[E_i]$ in $\widetilde{X}$, then this pulls back to a nontrivial linear relation among the pullback Cartier divisor classes on $\widehat{Y}$. By the argument above, the irreducible components of the exceptional locus on $\widehat{Y}$ are $\mathbb{Z}$-linearly independent. theater barth spielplanWebJan 10, 2024 · and well done. $\blacksquare$ Section 1.3. The Cone of Curves of Smooth Varieties. Definition 1.15. More properties of extremal faces and rays we refer chapter 18 (especially Theorem 18.5) in book [Convex97] 1 which is important for us to read the Mori’s theory. $\blacksquare$ Theorem 1.24. theater barrington nh