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In mathematics, the ELSV formula, named after its four authors Torsten Ekedahl, Sergei Lando, Michael Shapiro, Alek Vainshtein, is an equality between a Hurwitz number (counting ramified coverings of the sphere) and an integral over the moduli space of stable curves.

Several fundamental results in the intersection theory of moduli spaces of curves can be deduced from the ELSV formula, including the Witten conjecture, the Virasoro constraints, and the $\lambda _{g}$ -conjecture.

The formula

Define the Hurwitz number

h_{g;k_{1},\dots ,k_{n}}

as the number of ramified coverings of the complex projective line (Riemann sphere, P¹(C)) that are connected curves of genus g, with n numbered preimages of the point at infinity having multiplicities k₁, ..., k_n and m more simple branch points. Here if a covering has a nontrivial automorphism group G it should be counted with weight 1/|G|.

The ELSV formula then reads

h_{g;k_{1},\dots ,k_{n}}=m!\prod _{i=1}^{n}{\frac {k_{i}^{k_{i}}}{k_{i}!}}\int _{{\overline {\mathcal {M}}}_{g,n}}{\frac {c(E^{*})}{(1-k_{1}\psi _{1})\cdots (1-k_{n}\psi _{n})}}.

Here the notation is as follows:

g ≥ 0 is a nonnegative integer;
n ≥ 1 is a positive integer;
k₁, ..., k_n are positive integers;
$m=\sum k_{i}+n+2g-2$ ;
${\overline {\mathcal {M}}}_{g,n}$ is the moduli space of stable curves of genus g with n marked points;
E is the Hodge vector bundle and c(E*) the total Chern class of its dual vector bundle;
ψ_i is the first Chern class of the cotangent line bundle to the i-th marked point.

The numbers

h_{g;k_{1},\dots ,k_{n}}

in the left-hand side have a combinatorial definition and satisfy properties that can be proved combinatorially. Each of these properties translates into a statement on the integrals on the right-hand side of the ELSV formula Template:Harv.

The Hurwitz numbers

h_{g;k_{1},\dots ,k_{n}}

also have a definition in purely algebraic terms. With K = k₁ + ... + k_n and m = K + n + 2g − 2, let τ₁, ..., τ_m be transpositions in the symmetric group S_K and σ a permutation with n numbered cycles of lengths k₁, ..., k_n. Then

(\tau _{1},\dots ,\tau _{m},\sigma )

is a transitive factorization of identity of type (k₁, ..., k_n) if the product

\tau _{1}\cdots \tau _{m}\sigma

equals the identity permutation and the group generated by

\tau _{1},\dots ,\tau _{n}

is transitive.

Definition. $h_{g;k_{1},\dots ,k_{n}}$ is the number of transitive factorization of identity of type (k₁, ..., k_n) divided by K!

Example A. The number $h_{g;k}$ is 1/k! times the number of lists of transpositions $(\tau _{1},\dots ,\tau _{k+2g-1})$ whose product is a k-cycle. In other words, $h_{g;k}$ is 1/k times the number of factorizations of a given k-cycle into a product of k + 2g − 1 transpositions.

The equivalence between the two definitions of Hurwitz numbers (counting ramified coverings of the sphere, or counting transitive factorizations) is established by describing a ramified covering by its monodromy. More precisely: choose a base point on the sphere, number its preimages from 1 to K (this introduces a factor of K!, which explains the division by it), and consider the monodromies of the covering about the branch point. This leads to a transitive factorization.

The integral over the moduli space

The moduli space ${\overline {\mathcal {M}}}_{g,n}$ is a smooth Deligne–Mumford stack of (complex) dimension 3g − 3 + n. (Heuristically this behaves much like complex manifold, except that integrals of characteristic classes that are integers for manifolds are rational numbers for Deligne-Mumford stacks.)

The Hodge bundle E is the rank g vector bundle over the moduli space ${\overline {\mathcal {M}}}_{g,n}$ whose fiber over a curve (C, x₁, ..., x_n) with n marked points is the space of abelian differentials on C. Its Chern classes are denoted by

\lambda _{j}=c_{j}(E)\in H^{2j}({\overline {\mathcal {M}}}_{g,n},\mathbf {Q} ).

We have

c(E^{*})=1-\lambda _{1}+\lambda _{2}-\cdots +(-1)^{g}\lambda _{g}.

The ψ-classes. Introduce line bundles ${\mathcal {L}}_{1}$ , ..., ${\mathcal {L}}_{n}$ over ${\overline {\mathcal {M}}}_{g,n}$ . The fiber of ${\mathcal {L}}_{i}$ over a curve (C, x₁, ..., x_n) is the cotangent line to C at x_i. The first Chern class of ${\mathcal {L}}_{i}$ is denoted by

\psi _{i}=c_{1}({\mathcal {L}}_{i})\in H^{2}({\overline {\mathcal {M}}}_{g,n},\mathbf {Q} ).

The integrand. The fraction $1/(1-k_{i}\psi _{i})$ is interpreted as $1+k_{i}\psi _{i}+k_{i}^{2}\psi _{i}^{2}+\cdots$ , where the sum can be cut at degree 3g − 3 + n (the dimension of the moduli space). Thus the integrand is a product of n + 1 factors. We expand this product, extract from it the part of degree 3g − 3 + n and integrate it over the moduli space.

The integral as a polynomial. It follows that the integral

\int _{{\overline {\mathcal {M}}}_{g,n}}{\frac {c(E^{*})}{(1-k_{1}\psi _{1})\cdots (1-k_{n}\psi _{n})}}

is a symmetric polynomial in variables k₁, ..., k_n, whose monomials have degrees between 3g − 3 + n and 2g − 3 + n. The coefficient of the monomial $k_{1}^{d_{1}}\cdots k_{n}^{d_{n}}$ equals

\int _{{\overline {\mathcal {M}}}_{g,n}}(-1)^{j}\lambda _{j}\psi _{1}^{d_{1}}\cdots \psi _{n}^{d_{n}},

where $j=3g-3-n-\sum d_{i}$ .

Remark. The polynomiality of the numbers

{\frac {h_{g;k_{1},\dots ,k_{n}}}{m!}}\prod _{i=1}^{n}{\frac {k_{i}!}{k_{i}^{k_{i}}}}

was first conjectured by I. P. Goulden and D. M. Jackson. No proof independent from the ELSV formula is known.

Example B. Let g = n = 1. Then

\int _{{\overline {\mathcal {M}}}_{g,n}}{\frac {c(E^{*})}{(1-k_{1}\psi _{1})\cdots (1-k_{n}\psi _{n})}}=\int _{{\overline {\mathcal {M}}}_{1,1}}{\frac {1-\lambda _{1}}{1-k_{1}\psi _{1}}}=\left[\int _{{\overline {\mathcal {M}}}_{1,1}}\psi _{1}\right]k_{1}-\left[\int _{{\overline {\mathcal {M}}}_{1,1}}\lambda _{1}\right].

Example

Let n = g = 1. To simplify the notation, denote k₁ by k. We have m = K + n + 2g − 2 = k + 1.

According to Example B, the ELSV formula in this case reads

h_{1;k}=(k+1)!{\frac {k^{k}}{k!}}\int _{{\overline {\mathcal {M}}}_{1,1}}{\frac {1-\lambda _{1}}{1-k\psi _{1}}}=(k+1)k^{k}\left\{\left[\int _{{\overline {\mathcal {M}}}_{1,1}}\psi _{1}\right]k-\left[\int _{{\overline {\mathcal {M}}}_{1,1}}\lambda _{1}\right]\right\}.

On the other hand, according to Example A, the Hurwitz number h_{1, k} equals 1/k times the number of ways to decompose a k-cycle in the symmetric group S_k into a product of k + 1 transpositions. In particular, h_{1, 1} = 0 (since there are no transpositions in S₁), while h_{1, 2} = 1/2 (since there is a unique factorization of the transposition (1 2) in S₂ into a product of three transpositions).

Plugging these two values into the ELSV formula we find

\int _{{\overline {\mathcal {M}}}_{1,1}}\psi _{1}=\int _{{\overline {\mathcal {M}}}_{1,1}}\lambda _{1}={\frac {1}{24}}.

From which we deduce

h_{1;k}={\frac {(k^{2}-1)k^{k}}{24}}.

History

The ELSV formula was announced by Template:Harvtxt, but with an erroneous sign. Template:Harvtxt proved it for k₁ = ... = k_n = 1 (with the corrected sign). Template:Harvtxt proved the formula in full generality using the localization techniques. The proof announced by the four initial authors followed Template:Harv. Now that the space of stable maps to the projective line relative to a point has been constructed by Template:Harvtxt, a proof can be obtained immediately by applying the virtual localization to this space.

Template:Harvtxt, building on preceding work of several people, gave a unified way to deduce most known results in the intersection theory of ${\overline {\mathcal {M}}}_{g,n}$ from the ELSV formula.

Idea of proof

Let ${\mathcal {M}}_{g;k_{1},\dots ,k_{n}}$ be the space of stable maps f from a genus g curve to P¹(C) such that f has exactly n poles of orders $k_{1}\dots k_{n}$ .

The branching morphism br or the Lyashko-Looijenga map assigns to $f\in {\mathcal {M}}_{g;k_{1},\dots ,k_{n}}$ the unordered set of its m branch points in C with multiplicities taken into account. Actually, this definition only works if f is a smooth map. But it has a natural extension to the space of stable maps. For instance, the value of f on a node is considered a double branch point, as can be seen by looking at the family of curves C_t given by the equation xy = t and the family of maps f_t(x, y) = x + y. As t → 0, two branch points of f_t tend towards the value of f₀ at the node of C₀.

The branching morphism is of finite degree, but has infinite fibers. Our aim is now to compute its degree in two different ways.

The first way is to count the preimages of a generic point in the image. In other words, we count the ramified coverings of P¹(C) with a branch point of type (k₁, ..., k_n) at ∞ and m more fixec simple branch points. This is precisely the Hurwitz number $h_{g;k_{1},\dots ,k_{n}}$ .

The second way to find the degree of br is to look at the preimage of the most degenerate point, namely, to put all m branch points together at 0 in C.

The preimage of this point in ${\mathcal {M}}_{g;k_{1},\dots ,k_{n}}$ is an infinite fiber of br isomorphic to the moduli space ${\overline {\mathcal {M}}}_{g,n}$ . Indeed, given a stable curve with n marked points we send this curve to 0 in P¹(C) and attach to its marked points n rational components on which the stable map has the form $z\mapsto z^{k_{1}},\dots ,z\mapsto z^{k_{n}}$ . Thus we obtain all stable maps in ${\mathcal {M}}_{g;k_{1},\dots ,k_{n}}$ unramified outside 0 and ∞. Standard methods of algebraic geometry allow one to find the degree of a map by looking at an infinite fiber and its normal bundle. The result is expressed as an integral of certain characteristic classes over the infinite fiber. In our case this integral happens to be equal to the right-hand side of the ELSV formula.

Thus the ELSV formula expresses the equality between two ways to compute the degree of the branching morphism.

References

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Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules

Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.

A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running

The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more

There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang
One of the biggest reasons investing in a Singapore new launch is an effective things is as a result of it is doable to be lent massive quantities of money at very low interest rates that you should utilize to purchase it. Then, if property values continue to go up, then you'll get a really high return on funding (ROI). Simply make sure you purchase one of the higher properties, reminiscent of the ones at Fernvale the Riverbank or any Singapore landed property Get Earnings by means of Renting

In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang

Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules

Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.

A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running

The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more

There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang
One of the biggest reasons investing in a Singapore new launch is an effective things is as a result of it is doable to be lent massive quantities of money at very low interest rates that you should utilize to purchase it. Then, if property values continue to go up, then you'll get a really high return on funding (ROI). Simply make sure you purchase one of the higher properties, reminiscent of the ones at Fernvale the Riverbank or any Singapore landed property Get Earnings by means of Renting

In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang

Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules

Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.

A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running

The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more

There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang
One of the biggest reasons investing in a Singapore new launch is an effective things is as a result of it is doable to be lent massive quantities of money at very low interest rates that you should utilize to purchase it. Then, if property values continue to go up, then you'll get a really high return on funding (ROI). Simply make sure you purchase one of the higher properties, reminiscent of the ones at Fernvale the Riverbank or any Singapore landed property Get Earnings by means of Renting

In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang

Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules

Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.

A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running

The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more

There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang
One of the biggest reasons investing in a Singapore new launch is an effective things is as a result of it is doable to be lent massive quantities of money at very low interest rates that you should utilize to purchase it. Then, if property values continue to go up, then you'll get a really high return on funding (ROI). Simply make sure you purchase one of the higher properties, reminiscent of the ones at Fernvale the Riverbank or any Singapore landed property Get Earnings by means of Renting

In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang

Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules

Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.

A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running

The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more

There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang
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+In mathematics, the '''ELSV formula''', named after its four authors [[Torsten Ekedahl]], [[Sergei Lando]], [[Michael Shapiro (mathematician)|Michael Shapiro]], [[Alek Vainshtein]], is an equality between a [[Hurwitz number]] (counting [[ramified covering]]s of the sphere) and an integral over the [[moduli of algebraic curves|moduli space of stable curves]].
+Several fundamental results in the [[intersection theory]] of moduli spaces of curves can be deduced from the ELSV formula, including the [[Witten conjecture]], the [[Virasoro conjecture|Virasoro constraints]], and the [[Lambda g conjecture|<math>\lambda_g</math>-conjecture]].
+== The formula ==
+Define the ''Hurwitz number''
+:<math> h_{g; k_1, \dots, k_n} </math>
+as the number of ramified coverings of the complex projective line ([[Riemann sphere]], '''P'''<sup>1</sup>('''C''')) that are connected curves of genus ''g'', with ''n'' numbered preimages of the [[point at infinity]] having multiplicities ''k<sub>1</sub>'', ..., ''k<sub>n</sub>'' and ''m'' more simple [[branch point]]s. Here if a covering has a nontrivial automorphism group ''G'' it should be counted with weight 1/''|G|''.
+The ELSV formula then reads
+: <math>h_{g;k_1, \dots, k_n} = m! \prod_{i=1}^n \frac{k_i^{k_i}}{k_i!} \int_{\overline{\mathcal{M}}_{g,n}} \frac{c(E^*)}{(1-k_1\psi_1) \cdots (1-k_n \psi_n)}.</math>
+Here the notation is as follows:
+* ''g'' ≥ 0 is a nonnegative integer;
+* ''n'' ≥ 1 is a positive integer;
+* ''k<sub>1</sub>'', ..., ''k<sub>n</sub>'' are positive integers;
+* <math> m = \sum k_i + n + 2g-2 </math>;
+* <math> \overline{\mathcal{M}}_{g,n} </math> is the [[moduli space]] of [[stable curve]]s of genus ''g''  with ''n'' marked points;
+* ''E'' is the [[Hodge vector bundle]] and ''c(E*)'' the total [[Chern class]] of its dual vector bundle;
+* ψ<sub>''i''</sub> is the first Chern class of the cotangent line bundle to the ''i''-th marked point.
+The numbers
+:<math> h_{g;k_1, \dots, k_n} </math>
+in the left-hand side have a combinatorial definition and satisfy properties that can be proved combinatorially. Each of these properties translates into a statement on the  integrals on the right-hand side of the ELSV formula {{Harv|Kazarian|2009}}.
+== The Hurwitz numbers ==
+The Hurwitz numbers
+:<math> h_{g;k_1, \dots, k_n} </math>
+also have a definition in purely algebraic terms. With ''K'' = ''k<sub>1</sub>'' + ... + ''k<sub>n</sub>'' and ''m'' = ''K'' + ''n'' + 2''g'' − 2, let τ<sub>1</sub>, ..., τ<sub>''m''</sub> be transpositions in the symmetric group ''S''<sub>''K''</sub> and σ a permutation with ''n'' numbered cycles of lengths ''k<sub>1</sub>'', ..., ''k<sub>n</sub>''. Then
+:<math> ( \tau_1 , \dots, \tau_m, \sigma) </math>
+is a transitive factorization of identity of type ''(k<sub>1</sub>, ..., k<sub>n</sub>)'' if the product
+:<math> \tau_1 \cdots \tau_m \sigma </math>
+equals the identity permutation and the group generated by
+:<math>\tau_1, \dots, \tau_n</math>
+is [[transitive group|transitive]].
+'''Definition.''' <math> h_{g;k_1, \dots, k_n} </math> is the number of transitive factorization of identity of type ''(k<sub>1</sub>, ..., k<sub>n</sub>)'' divided by ''K''!
+'''Example A.''' The number <math> h_{g;k} </math> is 1/''k!'' times the number of lists of transpositions <math>(\tau_1, \dots, \tau_{k+2g-1})</math> whose product is a ''k''-cycle. In other words, <math>h_{g;k}</math> is 1/''k'' times the number of factorizations of a given ''k''-cycle into a product of ''k'' + 2''g'' − 1 transpositions.
+The equivalence between the two definitions of Hurwitz numbers (counting ramified coverings of the sphere, or counting transitive factorizations) is established by describing a ramified covering by its [[monodromy]]. More precisely: choose a base point on the sphere, number its preimages from 1 to ''K'' (this introduces a factor of ''K''!, which explains the division by it), and consider the monodromies of the covering about the branch point. This leads to a transitive factorization.
+== The integral over the moduli space ==
+'''The moduli space''' <math>{\overline {\mathcal{M}}}_{g,n}</math> is a smooth [[algebraic stack|Deligne–Mumford stack]] of (complex) dimension 3''g'' − 3 + ''n''. (Heuristically this behaves much like complex manifold, except that integrals of characteristic classes that are integers for manifolds are rational numbers for Deligne-Mumford stacks.)
+'''The Hodge bundle''' ''E'' is the rank ''g'' vector bundle over the moduli space <math>{\overline {\mathcal{M}}}_{g,n}</math> whose fiber over a curve ''(C, x<sub>1</sub>, ..., x<sub>n</sub>)'' with ''n'' marked points is the space of [[abelian differential]]s on ''C''. Its Chern classes are denoted by
+:<math>\lambda_j = c_j(E) \in H^{2j}({\overline {\mathcal{M}}}_{g,n},\mathbf{Q}). </math>
+We have
+:<math> c(E^*) = 1 - \lambda_1 + \lambda_2 - \cdots+ (-1)^g \lambda_g. </math>
+'''The ψ-classes.''' Introduce line bundles <math>\mathcal{L}_1</math>, ..., <math>\mathcal{L}_n</math> over <math>{\overline {\mathcal{M}}}_{g,n}</math>. The fiber of <math>\mathcal{L}_i</math> over a curve ''(C, x<sub>1</sub>, ..., x<sub>n</sub>)'' is the cotangent line to ''C'' at ''x<sub>i</sub>''. The first Chern class of <math>\mathcal{L}_i</math> is denoted by
+:<math>\psi_i = c_1(\mathcal{L}_i) \in H^2({\overline {\mathcal{M}}}_{g,n},\mathbf{Q}).</math>
+'''The integrand.''' The fraction <math> 1/(1-k_i \psi_i) </math> is interpreted as <math> 1 + k_i \psi_i + k_i^2 \psi_i^2 + \cdots </math>, where the sum can be cut at degree 3''g'' − 3 + ''n'' (the dimension of the moduli space). Thus the integrand is a product of ''n'' + 1 factors. We expand this product, extract from it the part of degree 3''g'' − 3 + ''n'' and integrate it over the moduli space.
+'''The integral as a polynomial.''' It follows that the integral
+:<math> \int_{\overline{\mathcal{M}}_{g,n}} \frac{c(E^*)}{(1-k_1\psi_1) \cdots (1-k_n \psi_n)} </math>
+is a symmetric polynomial in variables ''k<sub>1</sub>'', ..., ''k<sub>n</sub>'', whose monomials have degrees between 3''g'' − 3 + ''n'' and 2''g'' − 3 + ''n''. The coefficient of the monomial <math> k_1^{d_1} \cdots k_n^{d_n} </math> equals
+:<math> \int_{\overline{\mathcal{M}}_{g,n}} (-1)^j \lambda_j \psi_1^{d_1} \cdots \psi_n^{d_n}, </math>
+where <math>j = 3g-3-n - \sum d_i </math>.
+'''Remark.''' The polynomiality of the numbers
+:<math> \frac{h_{g;k_1, \dots, k_n}}{m!} \prod_{i=1}^n \frac{k_i!}{k_i^{k_i}}</math>
+was first conjectured by I. P. Goulden and D. M. Jackson. No proof independent from the ELSV formula is known.
+'''Example B.''' Let ''g'' = ''n'' = 1. Then
+:<math>\int_{\overline{\mathcal{M}}_{g,n}} \frac{c(E^*)}{(1-k_1\psi_1) \cdots (1-k_n \psi_n)}= \int_{\overline{\mathcal{M}}_{1,1}} \frac{1-\lambda_1}{1-k_1\psi_1} = \left[\int_{\overline{\mathcal{M}}_{1,1}} \psi_1 \right] k_1 - \left[ \int_{\overline{\mathcal{M}}_{1,1}} \lambda_1\right].</math>
+== Example ==
+Let ''n'' = ''g'' = 1.  To simplify the notation, denote ''k''<sub>1</sub> by ''k''. We have ''m'' = ''K'' + ''n'' + 2''g'' − 2 = ''k'' + 1.
+According to Example B, the ELSV formula in this case reads
+:<math>h_{1;k} = (k+1)!\frac{k^k}{k!} \int_{\overline{\mathcal{M}}_{1,1}} \frac{1-\lambda_1}{1-k\psi_1} = (k+1) k^k \left\{ \left[ \int_{\overline{\mathcal{M}}_{1,1}} \psi_1\right] k - \left[ \int_{\overline{\mathcal{M}}_{1,1}} \lambda_1 \right]\right\}. </math>
+On the other hand, according to Example A, the Hurwitz number ''h<sub>1, k</sub>'' equals 1/''k'' times the number of ways to decompose a ''k''-cycle in the symmetric group ''S<sub>k</sub>'' into a product of ''k'' + 1 transpositions. In particular, ''h<sub>1, 1</sub>'' = 0 (since there are no transpositions in ''S''<sub>1</sub>), while ''h<sub>1, 2</sub>'' = 1/2 (since there is a unique factorization of the transposition (1 2) in ''S<sub>2</sub>'' into a product of three transpositions).
+Plugging these two values into the ELSV formula we find
+:<math> \int_{\overline{\mathcal{M}}_{1,1}} \psi_1 = \int_{\overline{\mathcal{M}}_{1,1}} \lambda_1= \frac1{24}.</math>
+From which we deduce
+:<math>h_{1;k} = \frac{(k^2-1) k^k}{24}.</math>
+== History ==
+The ELSV formula was announced by {{Harvtxt|Ekedahl|Lando|Shapiro|Vainshtein|1999}}, but with an erroneous sign. {{Harvtxt|Fantechi|Pandharipande|2002}} proved it for ''k<sub>1</sub>'' = ... = ''k<sub>n</sub>'' = 1 (with the corrected sign). {{Harvtxt|Graber|Vakil|2003}} proved the formula in full generality using the localization techniques. The proof announced by the four initial authors followed {{Harv|Ekedahl|Lando|Shapiro|Vainshtein|2001}}. Now that the space of stable maps to the projective line relative to a point has been constructed by {{Harvtxt|Li|2001}}, a proof can be obtained immediately by applying the virtual localization to this space.
+{{Harvtxt|Kazarian|2009}}, building on preceding work of several people, gave a unified way to deduce most known results in the intersection theory of <math> {\overline{\mathcal{M}}}_{g,n} </math> from the ELSV formula.
+== Idea of proof ==
+Let <math> \mathcal{M}_{g;k_1, \dots, k_n} </math> be the space of stable maps ''f'' from a genus ''g'' curve to '''P'''<sup>1</sup>('''C''') such that ''f'' has exactly ''n'' poles of orders <math> k_1 \dots k_n </math>.
+The ''branching morphism'' ''br'' or the ''Lyashko-Looijenga map'' assigns to <math> f \in \mathcal{M}_{g;k_1, \dots, k_n} </math> the unordered set of its ''m'' branch points in '''C''' with multiplicities taken into account. Actually, this definition only works if ''f'' is a smooth map. But it has a natural extension to the space of stable maps. For instance, the value of ''f'' on a node is considered a double branch point, as can be seen by looking at the family of curves ''C<sub>t</sub>'' given by the equation ''xy'' = ''t'' and the family of maps ''f<sub>t</sub>(x, y)'' = ''x'' + ''y''. As ''t'' → 0, two branch points of ''f<sub>t</sub>'' tend towards the value of ''f<sub>0</sub>'' at the node of ''C<sub>0</sub>''.
+The branching morphism is of finite degree, but has infinite fibers. Our aim is now to compute its degree in two different ways.
+The first way is to count the preimages of a generic point in the image. In other words, we count the ramified coverings of '''P'''<sup>1</sup>('''C''') with a branch point of type ''(k<sub>1</sub>, ..., k<sub>n</sub>)'' at ∞ and ''m'' more fixec simple branch points. This is precisely the Hurwitz number <math> h_{g;k_1, \dots, k_n} </math>.
+The second way to find the degree of ''br'' is to look at the preimage of the most degenerate point, namely, to put all ''m'' branch points together at 0 in '''C'''.
+The preimage of this point in <math>\mathcal{M}_{g;k_1, \dots, k_n}</math> is an infinite fiber of ''br'' isomorphic to the moduli space <math>\overline{\mathcal{M}}_{g,n} </math>. Indeed, given a stable curve with ''n'' marked points we send this curve to 0 in '''P'''<sup>1</sup>('''C''') and attach to its marked points ''n'' rational components on which the stable map has the form <math> z \mapsto z^{k_1}, \dots, z \mapsto z^{k_n} </math>. Thus we obtain all stable maps in <math> \mathcal{M}_{g;k_1, \dots, k_n} </math> unramified outside 0 and ∞. Standard methods of algebraic geometry allow one to find the degree of a map by looking at an infinite fiber and its normal bundle. The result is expressed as an integral of certain characteristic classes over the infinite fiber. In our case this integral happens to be equal to the right-hand side of the ELSV formula.
+Thus the ELSV formula expresses the equality between two ways to compute the degree of the branching morphism.
+== References ==
+<!--- See [[Wikipedia:Footnotes]] on how to create references using <ref></ref> tags which will then appear here automatically -->
+*{{cite journal |ref=harv |id={{ArXiv|math|9902104}} |first=T. |last=Ekedahl |first2=S. |last2=Lando |first3=M. |last3=Shapiro |first4=A. |last4=Vainshtein |title=On Hurwitz numbers and Hodge integrals |journal=Comptes Rendus de l'Académie des Sciences |volume=328 |issue=12 |year=1999 |pages=1175–1180 |doi=10.1016/S0764-4442(99)80435-2 }}
+*{{cite journal |ref=harv |first=T. |last=Ekedahl |first2=S. |last2=Lando |first3=M. |last3=Shapiro |first4=A. |last4=Vainshtein |id={{ArXiv|math|0004096}} |title=Hurwitz numbers and intersections on moduli spaces of curves |journal=Invent. Math. |volume=146 |issue=2 |pages=297–327 |year=2001 |doi= }}
+*{{cite journal |ref=harv |first=B. |last=Fantechi |first2=R. |last2=Pandharipande |id={{ArXiv|math|9905104}} |title=Stable maps and branch divisors |journal=Compos. Math. |volume=130 |issue=3 |pages=345–364 |year=2002 }}
+*{{cite journal |ref=harv |first=T. |last=Graber |first2=R. |last2=Vakil |id={{ArXiv|math|0003028}} |title=Hodge integrals and Hurwitz numbers via virtual localization |journal=Compos. Math. |volume=135 |issue=1 |pages=25–36 |year=2003 |doi= }}
+*{{cite journal |ref=harv |first=M. |last=Kazarian |arxiv=0809.3263 |title=KP hierarchy for Hodge integrals |journal=Adv. Math. |volume=221 |issue=1 |pages=1–21 |year=2009 |doi= }}
+*{{cite paper |ref=harv |first=J. |last=Li |year=2001 |id={{ArXiv|math|0009097}} |title=A degeneration of stable morphisms and relative stable morphisms |work=Preprint }}
+{{Algebraic curves navbox}}
+[[Category:Algebraic curves]]
+[[Category:Moduli theory]]
+[[Category:Permutations]]

David Mount: Difference between revisions

Revision as of 10:41, 24 December 2013

Contents

The formula

The Hurwitz numbers

The integral over the moduli space

Example

History

Idea of proof

References

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David Mount: Difference between revisions

Revision as of 10:41, 24 December 2013

The formula

The Hurwitz numbers

The integral over the moduli space

Example

History

Idea of proof

References

Navigation menu

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