Divisor-- line package document in algebraic geometry

I recognize a little of the concept of portable Riemann surface areas, in which there is a really wonderful divisor-- line package document.

Yet when I occupy guide of Hartshorne, the idea of Cartier divisor there is really complex. It is absolutely not a straight amount of factors; probably it is far better to recognize it in regards to line packages. Yet Cartier divisors do not appear to be fairly the very same point as line packages. The definition is tough to identify. Can a person clear the misconception for me and also clarify to me just how ideal to recognize Cartier divisors?

2019-05-07 15:04:07
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When reviewing divisors, a handy difference to make at the start works divisors vs. all divisors. Generally reliable divisors have an even more geometric summary ; all divisors can after that be gotten from the reliable ones by permitting some minus indicators ahead right into the image.

An irreducible reliable Weil divisor on a selection $X$ coincides point as an irreducible codimension one subvariety, which subsequently coincides point as an elevation one factor $\eta$ of $X$. (We get $\eta$ as the common factor of the irred. codim 'n one subvariety, and also we recoup the subvariety as the closure of $\eta$.)
A reliable Weil divisor is a non - adverse indispensable straight mix of irreducible ones, so you can consider it as a non - adverse indispensable straight mix of elevation one aims $\eta$.

Commonly, one limits to regular selections, to make sure that all the neighborhood rings at elevation one factors are DVRs. After that, offered any kind of pure codimension one subscheme $Z$ of $X$, you can attach a Weil divisor to $Z$, in the list below means : due to the fact that the neighborhood rings at elevation one factors are DVRs, if $Z$ is any kind of codimension one subscheme of $X$, removed by an excellent sheaf $\mathcal I_Z$, and also $\eta$ is an elevation one factor, after that the stalk $\mathcal I_{Z,\eta}$ is a perfect in the DVR $\mathcal O_{X,\eta}$, hence is simply some power of the topmost perfect $\mathfrak m_{\eta}$ (making use of the DVR building), claim $\mathcal I_{Z,\eta} = \mathfrak m_{\eta}^{m_{Z,\eta}},$ therefore the multiplicity $m$ of $Z$ at $\eta$ is well - specified.
Hence the reliable Weil divisor $$div(Z) := \sum_{\eta \text{ of height one}} m_{Z,\eta}\cdot \eta$$ is well - specified.

Keep in mind that this dish just goes one means : beginning with the Weil divisor, we can not recoup $Z$, due to the fact that the Weil divisor does not bear in mind all the system framework (i.e. the entire framework sheaf, or equivalently, the entire excellent sheaf) of $Z$, yet just its practices at its common factors (which totals up to the very same point as bearing in mind the irreducible parts and also their multiplicities).

A reliable Cartier divisor is in fact an extra straight geometric object, particularly, it is a in your area major pure codimension one subscheme, that is, a subscheme, each part of which is codimension one, and also which, in your area around each factor, is the absolutely no locus of an area of the framework sheaf. Currently in order to remove a pure codimension one subscheme as its absolutely no locus, an area of the framework sheaf needs to be normal (in the commutative algebra feeling), i.e. a non - absolutely no divisor. Additionally, 2 normal areas will certainly remove the very same absolutely no locus if their proportion is a device in the framework sheaf. So if we allowed $\mathcal O_X^{reg}$ represent the subsheaf of $\mathcal O_X$ whose areas are normal components (i.e. non - absolutely no divisors in each stalk),. after that the formula of a Cartier divisor is a well - specified international area of the ratio. sheaf. $\mathcal O_X^{reg}/\mathcal O_X^{\times}$.

Currently intend that we get on a smooth selection. After that any kind of irreducible codimension one subvariety. remains in reality in your area major, therefore offered a Weil divisor $$D = \sum_{\eta \text{ of height one}} m_{\eta} \cdot\eta,$$. we can in fact canonically attach a Cartier divisor to it, in the list below means : in a n.h. of some factor $x$, allow $f_{\eta}$ be a neighborhood formula for the Zariski closure. of $\eta$ ; after that if $Z(D)$ is removed in your area by $\prod_{\eta} f_{\eta}^{m_{\eta}} = 0,$. after that $Z(D)$ is in your area major by building and construction, and also, once more by building and construction,. $div(Z(D)) = D.$

So in the smooth setup,. we see that $Z \mapsto div(Z)$ and also $D \mapsto Z(D)$ develop a bijection in between. reliable Cartier divisors and also reliable Weil divisors.

On the various other hand, on a single selection, it can take place that an irreducible codimension one subvariety need not be in your area major in the area of a single factor (as an example a creating line on the cone $x^2 +y^2 + z^2 = 0$. in $\mathbb A^3$ is not in your area. principal in any kind of area of the cone factor). Hence there can be Weil divisors that. are not of the kind $div(Z)$ for any kind of Cartier divisor $Z$.

To go from reliable Weil divisor to all Weil divisors, you simply permit adverse coefficients.

To go from reliable Cartier divisors to all Cartier divisors, you need to permit on your own to invert the features $f$ that removed the reliable Cartier divisors, or equivalently, to go from the sheaf of monoids $\mathcal O_X^{reg}/\mathcal O_X^{\times}$ to the linked sheaf of teams, which is $\mathcal K_X^{reg}/\mathcal O_X^{\times}.$ (Here, it aids to bear in mind that $\mathcal K_X$ is gotten from $\mathcal O_X$. by inverting non - absolutely no divisors.)

Ultimately, for the link with line packages : if $\mathcal L$ is a line package,. and also $s$ is a normal area (i.e. an area whose absolutely no locus is pure codimension one,. or equivalently, an area which, when we pick a neighborhood isomorphism $\ mathcal L _ U \ cong \ mathcal O_U$, is not a zero divisor), then the zero locus $Z (s) $ of $s$. is a reliable Cartier divisor, basically necessarily.

So we have a map $(\mathcal L,s) \mapsto Z(s)$ which sends out line packages with normal areas to reliable Cartier divisors. This remains in reality an isomorphism of monoids (where left wing we take into consideration sets $(\mathcal L,s)$ approximately isomorphism of sets) : offered a reliable Cartier divisor $D$, we can specify $\mathcal O(D)$ to be the. subsheaf of $\mathcal K_X$ being composed (in your area) of areas $f$ such that the locus of. posts of $f$ (a well - specified Cartier divisor) is had (as a subscheme) in the Cartier. divisor $D$ (probably much less with ease, yet extra concretely : if $D$ is in your area removed. by the formula $g = 0$, after that $\mathcal O(D)$ is composed (in your area) of areas $f$. of $\mathcal K_X$ such that $fg$ remains in reality an area of $\mathcal O_X$).

The constant function $1$ absolutely hinges on $\mathcal O(D)$, and also (idea of as an area. of $\mathcal O(D)$ - - not as a function!) its absolutely no locus is specifically $D$.

Hence $D \mapsto (\mathcal O(D), 1)$ is an inverted to the above map $ (\ mathcal L, s) \ mapsto. Z (s) $.

Ultimately, if we pick 2 various normal areas of the very same line package,. the equivalent Cartier divisors are linearly equal. Hence we are brought about the. isomorphism "line dress to isomorphism = Cartier divisors approximately straight equivalence". Yet, simply to stress, to recognize this it is best to limit first to line packages. which confess a normal area, and afterwards consider the equivalent Cartier divisor as being. the absolutely no locus of that area. This highlights the geometric nature of the Cartier divisor fairly plainly.

2019-05-09 10:25:17