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In algebraic geometry, a linear system of divisors is an algebraic generalization of the geometric notion of a family of curves; the dimension of the linear system corresponds to the number of parameters of the family.

These arose first in the form of a linear system of algebraic curves in the projective plane. It assumed a more general form, through gradual generalisation, so that one could speak of linear equivalence of divisors D on a general scheme or even a ringed space (X, O_X).^[1]

A linear system of dimension 1, 2, or 3 is called a pencil, a net, or a web.

Definition by means of functions

Given the fundamental idea of a rational function on a general variety V, or in other words of a function f in the function field of V, divisors D and E are linearly equivalent if

D=E+(f)\

where (f) denotes the divisor of zeroes and poles of the function f.

Note that if V has singular points, 'divisor' is inherently ambiguous (Cartier divisors, Weil divisors: see divisor (algebraic geometry)). The definition in that case is usually said with greater care (using invertible sheaves or holomorphic line bundles); see below.

A complete linear system on V is defined as the set of all effective divisors linearly equivalent to some given divisor D. It is denoted |D|. Let L(D) be the line bundle associated to D. It can be proved that |D| corresponds bijectively to $(\Gamma (V,L)\setminus \{0\})/k^{\ast },$ ^[2] and is therefore a projective space.

A linear system ${\mathfrak {d}}$ is then a projective subspace of a complete linear system, so it corresponds to a vector subspace W of $\Gamma (V,L).$ The dimension of the linear system ${\mathfrak {d}}$ is its dimension as a projective space. Hence $\dim {\mathfrak {d}}=\dim W-1$ .

Base locus

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i) registered as a patent agent or its equal in any nation or territory, or by a patent workplace, specified within the Fourth Schedule; The business-specific tips for the true property agency and telecommunication sectors have been crafted to address considerations about scenarios that particularly apply to the two sectors, the PDPC stated. Mr Steven Tan, Managing Director of OrangeTee real property company, nonetheless, felt that it was a matter of "practising until it becomes part of our knowledge". "After a while, the agents ought to know the spirit behind the (Act)," he stated. Rising office sector leads real property market efficiency, while prime retail and enterprise park segments moderate and residential sector continues in decline Please choose an attendee for donation. If all divisors in the system share common points, this is referred to as the base locus of the linear system. Geometrically, this corresponds to the common intersection of the varieties. Linear systems may or may not have a base locus – for example, the pencil of affine lines $x=a$ has no common intersection, but given two (nondegenerate) conics in the complex projective plane, they intersect in four points (counting with multiplicity) and thus the pencil they define has these points as base locus.

Linear system of conics

For example, the conic sections in the projective plane form a linear system of dimension five, as one sees by counting the constants in the degree two equations. The condition to pass through a given point P imposes a single linear condition, so that conics C through P form a linear system of dimension 4. Other types of condition that are of interest include tangency to a given line L.

In the most elementary treatments a linear system appears in the form of equations

\lambda C+\mu C'=0\

with λ and μ unknown scalars, not both zero. Here C and C′ are given conics. Abstractly we can say that this is a projective line in the space of all conics, on which we take

[\lambda :\mu ]\

as homogeneous coordinates. Geometrically we notice that any point Q common to C and C′ is also on each of the conics of the linear system. According to Bézout's theorem C and C′ will intersect in four points (if counted correctly). Assuming these are in general position, i.e. four distinct intersections, we get another interpretation of the linear system as the conics passing through the four given points (note that the codimension four here matches the dimension, one, in the five-dimensional space of conics). Note that of these conics, exactly three are degenerate, each consisting of a pair of lines, corresponding to the $\textstyle {{\binom {4}{2,2}}/2=3}$ ways of choosing 2 pairs of points from 4 points (counting via the multinomial coefficient, and accounting for the overcount by a factor of 2 that $\textstyle {\binom {4}{2}}$ makes when interested in counting pairs of pairs rather than just selections of size 2).

Applications

A striking application of such a family is in Template:Harv which gives a geometric solution to a quartic equation by considering the pencil of conics through the four roots of the quartic, and identifying the three degenerate conics with the three roots of the resolvent cubic.

Example

Template:External media For example, given the four points $(\pm 1,\pm 1),$ the pencil of conics through them can be parameterized as $ax^{2}+(1-a)y^{2}=1,$ which are the affine combinations of the equations $x^{2}=1$ and $y^{2}=1,$ corresponding to the parallel vertical lines and horizontal lines; this yields degenerate conics at the standard points of $0,1,\infty .$ A less elegant but more symmetric parametrization is given by $(1+a)x^{2}+(1-a)y^{2}=2,$ in which case inverting a ( $a\mapsto -a$ ) interchanges x and y, yielding the following pencil; in all cases the center is at the origin:

$a>1:$ hyperbolae opening left and right;
$a=1:$ the parallel vertical lines $x=-1,x=1;$

(intersection point at [1:0:0])

$0<a<1:$ ellipses with a vertical major axis;
$a=0:$ a circle (with radius ${\sqrt {2}}$ );
$-1<a<0:$ ellipses with a horizontal major axis;
$a=-1:$ the parallel horizontal lines $y=-1,y=1;$

(intersection point at [0:1:0])

$a<-1:$ hyperbolae opening up and down,
$a=\infty :$ the diagonal lines $y=x,y=-x;$

(dividing by

a

and taking the limit as

a\to \infty

yields

x^{2}-y^{2}=0

)

(intersection point at [0:0:1])

This then loops around to $a>1,$ since pencils are a projective line.

In the terminology of Template:Harv, this is a Type I linear system of conics, and is animated in the linked video.

Classification

There are 8 types of linear systems of conics over the complex numbers, depending on intersection multiplicity at the base points, which divide into 13 types over the real numbers, depending on whether the base points are real or imaginary; this is discussed in Template:Harv and illustrated in Template:Harv.

Other examples

The Cayley–Bacharach theorem is a property of a pencil of cubics, which states that the base locus satisfies an "8 implies 9" property: any cubic containing 8 of the points necessarily contains the 9th.

Linear systems in birational geometry

In general linear systems became a basic tool of birational geometry as practised by the Italian school of algebraic geometry. The technical demands became quite stringent; later developments clarified a number of issues. The computation of the relevant dimensions — the Riemann-Roch problem as it can be called — can be better phrased in terms of homological algebra. The effect of working on varieties with singular points is to show up a difference between Weil divisors (in the free abelian group generated by codimension-one subvarieties), and Cartier divisors coming from sections of invertible sheaves.

The Italian school liked to reduce the geometry on an algebraic surface to that of linear systems cut out by surfaces in three-space; Zariski wrote his celebrated book Algebraic Surfaces to try to pull together the methods, involving linear systems with fixed base points. There was a controversy, one of the final issues in the conflict between 'old' and 'new' points of view in algebraic geometry, over Henri Poincaré's characteristic linear system of an algebraic family of curves on an algebraic surface.

Line bundle/invertible sheaf language

Linear systems are still at the heart of contemporary algebraic geometry; but they are typically introduced by means of the line bundle or invertible sheaf language. That is, the language of sheaf theory is considered the most natural starting point, at least to learn the theory. In those terms, divisors D (Cartier divisors, importantly) correspond to line bundles, and linear equivalence of two divisors means that the corresponding line bundles are isomorphic.

References

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15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010
Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010
20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.

My blog: http://www.primaboinca.com/view_profile.php?userid=5889534
Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010

Template:Refend

↑ Grothendieck, Alexandre; Dieudonné, Jean. EGA IV, 21.3.
↑ Hartshorne, R. 'Algebraic Geometry', proposition II.7.7, page 157

[1] Grothendieck, Alexandre; Dieudonné, Jean. EGA IV, 21.3.

[2] Hartshorne, R. 'Algebraic Geometry', proposition II.7.7, page 157

[1]

[2]