Stein factorization. ([107], [241])

Theorem. (“algebraic version”; see [107]) Let 𝑓 : 𝑋 → 𝑌 be a projective morphism of Noetherian schemes (see“Schemes”). Then there exists a scheme 𝑍, a finite morphism 𝑔 : 𝑍 → 𝑌 and a projective morphism with connected fibres ℎ : 𝑋 → 𝑍 such that

𝑓 = 𝑔 ∘ ℎ .

Theorem. (“analytic version”); (see [241]). Let 𝑓 : 𝑋 → 𝑌 be a proper surjective mor-phism of reduced complex analytic spaces (see“Spaces, analytic -”). Then there exists a reduced complex analytic space 𝑍, a surjective morphism 𝑔 : 𝑍 → 𝑌 such that the fibres consist of a finite number of points and a surjective morphism with connected fibres ℎ : 𝑋 → 𝑍 such that

𝑓 = 𝑔 ∘ ℎ .

Subcanonical.

We say that a smooth subvariety 𝑍 of a smooth algebraic variety 𝑋 is subcanonical if there exists a line bundle 𝐿 on 𝑋 such that 𝐿|_𝑍 = 𝐾_𝑍, where 𝐾_𝑍is the canonical bundle of 𝑍 (see“Canonical bundle, canonical sheaf”).

Surfaces, algebraic -.

([18], [22], [25], [65], [93], [107], [253]). In the sequel the word surface will denote a smooth projective algebraic variety of dimension 2 over ℂ.

We will denote the projective space ℙ^𝑛_ℂby ℙ^𝑛.

We start by stating some basic theorems and notation.

Let 𝑆 be a surface; then 𝐻4(𝑆, ℤ) and 𝐻⁴(𝑆, ℤ) are isomorphic to ℤ, since 𝑆 is oriented (as it is a complex manifold), compact and of real dimension 4 (see“Singular homol-ogy and cohomolhomol-ogy”). We denote by [𝑆] the fundamental class of 𝑆, that is, the gener-ator of 𝐻₄(𝑆, ℤ) giving the orientation of 𝑆. If 𝐿 and 𝐿^󸀠are two holomorphic line bundles on 𝑆, we denote by 𝐿 ⋅ 𝐿^󸀠the number obtained by evaluating 𝑐1(𝐿) ∪ 𝑐₁(𝐿^󸀠) ∈ 𝐻⁴(𝑆, ℤ) in [𝑆] (where ∪ is the cup product; see again“Singular homology and cohomology”).

For any divisors 𝐷 and 𝐷^󸀠on 𝑆, let 𝐷 ⋅ 𝐷^󸀠their intersection number (see“Intersection of cycles”).

Observe that, for any holomorphic line bundle 𝐿 on 𝑆, we have that 𝐻^𝑖(^O(𝐿)) = 0 for 𝑖 ≥ 3; to prove this, we can apply the Abstract de Rham’s theorem (see“Sheaves”) to the exact sequence

0 󳨀→^O(𝐿) 󳨀→ 𝐶^∞(𝐿)󳨀→ 𝐴^{𝜕 0,1}(𝐿)󳨀→ 𝐴^{𝜕 0,2}(𝐿) 󳨀→ 0,

where 𝐶^∞(𝐿) is the sheaf of the 𝐶^∞sections of 𝐿, 𝐴^𝑝,𝑞(𝐿) is the sheaf of the 𝐶^∞(𝑝, 𝑞)-forms with values in 𝐿 and the map^O(𝐿) 󳨀→ 𝐶^∞(𝐿) is given by inclusion.

So, if we denote by 𝜒(^O(𝐿)) the Euler characteristic of ^O(𝐿), i.e., the number

∑_{𝑖=0,...,∞}(−1)^𝑖ℎ^𝑖(^O(𝐿)), we have that, in this case (i.e., in the case of surfaces), 𝜒(^O(𝐿)) = ℎ⁰(^O(𝐿)) − ℎ¹(^O(𝐿)) + ℎ²(^O(𝐿)),

Surfaces, algebraic - | 181 and so 𝜒(^O𝑆) = ℎ⁰(^O_𝑆) − ℎ¹(^O_𝑆) + ℎ²(^O_𝑆). Furthermore, we recall that 𝜒(𝑆) denotes the Euler-Poincaré characteristic of 𝑆, i.e., ∑_{𝑖=0,...,∞}(−1)^𝑖𝑏_𝑖(𝑆), where 𝑏_𝑖(𝑆) is the Betti number 𝑑𝑖𝑚 𝐻𝑖(𝑆, ℝ); in this case,

𝜒(𝑆) = 𝑏₀(𝑆) − 𝑏₁(𝑆) + 𝑏₂(𝑆) − 𝑏₃(𝑆) + 𝑏₄(𝑆) = 2𝑏₀(𝑆) − 2𝑏₁(𝑆) + 𝑏₂(𝑆) by Poincaré duality (see“Singular homology and cohomology”).

Index theorem. Let 𝑆 be a surface. The intersection form is negative definite on a sub-space of codimension 1 in 𝐻^1,1(𝑆, ℂ), precisely is negative definite on the subspace of the primitive forms (see“Lefschetz decomposition and Hard Lefschetz theorem”for the definition of primitive).

Riemann–Roch theorem for surfaces. Let 𝐿 be a holomorphic line bundle on a sur-face 𝑆. We have:

𝜒(^O(𝐿)) = 𝜒(^O_𝑆) +𝐿 ⋅ 𝐿 − 𝐿 ⋅ 𝐾_𝑆

2 ,

where 𝐾_𝑆 is the canonical bundle of 𝑆 (see “Canonical bundle, canonical sheaf”).

Noether’s theorem. Let 𝑆 be a surface. We denote by 𝑐_𝑖(𝑆) the 𝑖-th Chern class of the holomorphic tangent bundle 𝑇^1,0𝑆 and we denote by 𝑐₁(𝑆)²the cup product 𝑐₁(𝑆)∪𝑐₁(𝑆).

Hopf theorem (see“Gauss–Bonnet–Hopf theorem”)𝑐₂(𝑇^1,0𝑆) = 𝜒(𝑆).

Castelnuovo–Enriques criterion. Let 𝑆 be a surface and 𝐶 a smooth rational curve in 𝑆 with 𝐶 ⋅ 𝐶 = −1. Then there exists a surface 𝑆^󸀠and a map 𝑝 : 𝑆 → 𝑆^󸀠such that 𝑝 is the blow-up of 𝑆^󸀠in a point 𝑃 and 𝑝⁻¹(𝑃) = 𝐶.

Structure of birational maps on surfaces. Let 𝑆₁and 𝑆₂be two surfaces and 𝑓 : 𝑆₁→ 𝑆₂be a birational map. Then there exists another surface ̃𝑆 and two morphisms, 𝑔₁ and 𝑔₂, both given by a sequence of blowing-up maps, such that the following diagram commutes:

182 | Surfaces, algebraic

-In other words, a birational map between surfaces is given by a sequence of blowing-ups followed by a sequence of blowing-downs.

Definition. We say that a surface 𝑆 is minimal if there does not exist another surface

̃𝑆 and a blowing-up 𝑆 → ̃𝑆. A minimal model for a surface 𝑆 is a minimal surface bira-tional to 𝑆.

We recall that if 𝑀 is a compact complex manifold of dimension 𝑛, we define – 𝑞(𝑀) = ℎ^1,0(𝑀) (irregularity);

– 𝑃_𝑟(𝑀) = ℎ⁰(^O(𝐾_𝑀^𝑟 )) for any 𝑟 ∈ ℕ (plurigenera);

– 𝑝𝑔(𝑀) = ℎ^𝑛,0(𝑀) = 𝑃₁(𝑀) (geometric genus).

One can prove that they are birational invariants. With the notation above, we have 𝜒(^O_𝑆) = ℎ⁰(^O_𝑆) − ℎ¹(^O_𝑆) + ℎ²(^O_𝑆)

= ℎ^0,0(𝑆) − ℎ^0,1(𝑆) + ℎ^0,2(𝑆) = 1 − 𝑞(𝑆) + 𝑝_𝑔(𝑆) by Dolbeault’s theorem and the Hodge theorem (see“Hodge theory”).

The most important tool to classify surfaces is Kodaira dimension 𝐾 (see“Kodaira dimension (or Kodaira number)”). For surfaces, it can be obviously only −∞, 0, 1, 2.

First, we will deal with rational and ruled surfaces.

Definition. We say that a surface is rational if it is birational to ℙ².

Definition. We say that a surface is ruled over a compact Riemann surface 𝐶 if it is birational to 𝐶 × ℙ¹.

Remark. Let 𝑆 be a surface. Then 𝑆 is ruled on ℙ¹⇔ 𝑆 is rational (in fact it is birational to ℙ¹× ℙ¹if and only if it is birational to ℙ²).

Definition. We say that a surface 𝑆 is geometrically ruled over a compact Riemann surface 𝐶 if there exists a smooth morphism 𝑆 → 𝐶 such that the fibres are isomorphic to ℙ¹.

(Please note that in some works the term “ruled” means “geometrically ruled”.) Noether–Enriques theorem. A geometrically ruled surface is equal to the projectivized ℙ(𝐸) of a vector bundle 𝐸 of rank 2.

Proposition 1. Let 𝑆 be a ruled surface on a compact Riemann surface 𝐶. Let 𝑔(𝐶) be the genus of 𝐶. Then

𝑞(𝑆) = 𝑔(𝐶), 𝑝_𝑔(𝑆) = 0, 𝑃_𝑛(𝑆) = 0 ∀𝑛 ≥ 2.

Moreover, if 𝑆 is geometrically ruled, then

𝐾_𝑆²= 8(−𝑔(𝐶) + 1), 𝑏₂(𝑆) = 2 .

Surfaces, algebraic - | 183

Definition. Let 𝑛 ∈ ℕ. The 𝑛-th Hirzebruch surface is 𝑆_𝑛= ℙ(^O_ℙ¹(𝑛) ⊕^O_ℙ¹),

where^O_ℙ¹is the trivial bundle on ℙ¹and^O_ℙ¹(𝑛) is the 𝑛-th power of the hyperplane bundle on ℙ¹; see“Hyperplane bundles, twisting sheaves”(we are making a slight abuse of the notation:^O_ℙ¹(𝑛) generally denotes the sheaf of the holomorphic sections of the 𝑛-th power of the hyperplane bundle, but sometimes, as here, it denotes just the 𝑛-th power of the hyperplane bundle).

Remark. The Hirzebruch surfaces are exactly the surfaces that are geometrically ruled on ℙ¹. In fact, by the Grothendieck–Segre theorem (see “Grothendieck–Segre theo-rem”) and by the Noether–Enriques theorem, we have that a surface that is geomet-rically ruled on ℙ¹is equal to ℙ(𝐿₁⊕ 𝐿₂) = ℙ((𝐿₁⊗ 𝐿⁻¹₂ ) ⊕^O_ℙ¹) for some 𝐿₁and 𝐿₂ holomorphic line bundles on ℙ¹and ℙ((𝐿₁⊗ 𝐿⁻¹₂ ) ⊕^O_ℙ¹) = ℙ(^O_ℙ¹(𝑛) ⊕^O_ℙ¹) for some 𝑛. Obviously we can suppose 𝑛 ∈ ℕ.

We can prove that for all 𝑛 ̸= 0 the surface 𝑆_𝑛is the unique ℙ¹-bundle on ℙ¹with an irreducible curve 𝐸 with 𝐸 ⋅ 𝐸 = −𝑛 and that the blow-up of ℙ²in a point is 𝑆₁. We can also prove that 𝑆_𝑛−1can be obtained from 𝑆_𝑛by a up followed by a blowing-down.

Let 𝐹 be a fibre of ℙ(^O_ℙ¹(𝑛) ⊕^O_P¹) and 𝑍 be the image of the section given by the zero section of^O_ℙ¹(𝑛) and by the section 1 of^O_P¹. Let 𝑆_𝑘,𝑛be the image of 𝑆_𝑛by the map 𝜑_{(𝑍+𝑘𝐹)}associated to the linear system |𝑍 + 𝑘𝐹| (see“Bundles fibre -”). We can prove that it is a surface in ℙ^𝑛+2𝑘+1of degree 𝑛+2𝑘, which is the minimal achievable degree for a surface in ℙ^𝑛+2𝑘+1(see“Minimal degree”). The surfaces 𝑆_𝑘,𝑛are the rational normal scrolls of dimension 2 (see“Scrolls, rational normal -”).

Proposition 2. Every nondegenerate surface of minimal degree in ℙ^𝑑, i.e., every non-degenerate surface of degree 𝑑 − 1 in ℙ^𝑑, is either a rational normal scroll or the Veronese surface in ℙ⁵, i.e., the image of ℙ²embedded in ℙ⁵by^O_ℙ2(2) (see“Veronese embedding”).

Another example of rational surfaces are Del Pezzo surfaces. Let 𝑟 ≤ 6. Let 𝑃₁, . . . , 𝑃_𝑟 be 𝑟 distinct points in ℙ²in general position (i.e., no 3 of them are collinear, and no 6 of them lie on a conic). Let 𝜋 : 𝑆 → ℙ²be the blowing-up of ℙ²in 𝑃₁, . . . , 𝑃_𝑟. Then

−𝐾_𝑆defines an embedding 𝑖 : 𝑆 󳨅→ ℙ^9−𝑟, whose image has degree 9 − 𝑟 and is called a Del Pezzo surface.

If 𝑟 = 6, 𝑖(𝑆) is a smooth cubic in ℙ³; if 𝑟 = 5, 𝑖(𝑆) is a complete intersection of two quadrics in ℙ⁴.

One can show that 𝑖(𝑆) contains only a finite number of lines, precisely the images under 𝑖 of

(a) the exceptional curves;

(b) the strict transforms of the lines 𝑃_𝑖𝑃_𝑗for 𝑖 ̸= 𝑗;

184 | Surfaces, algebraic

-(c) the strict transforms of the conics through 5 of the 𝑃𝑖. In the case 𝑟 = 6 we have exactly 27 lines.

Theorem.

(i) The minimal ruled surfaces on ℙ¹are isomorphic either to the Hirzebruch surfaces or to ℙ².

(ii) The minimal ruled surfaces on a Riemann surface of genus ≥ 1 are the geometri-cally ruled ones and the minimal model is not unique.

Theorem. The nonruled surfaces have a unique minimal model (up to isomor-phisms).

Remark. If 𝑆 is a rational surface, then, for every 𝑛 ≥ 1, we have 𝑃_𝑛(𝑆) = 𝑞(𝑆) = 𝑝_𝑔(𝑆) = 0.

The following theorem tells us that also the converse is true.

Castelnuovo–Enriques theorem. Let 𝑆 be a surface such that 𝑞(𝑆) = 𝑃₂(𝑆) = 0. Then 𝑆 is rational.

Theorem. Any unirational surface (see “Unirational, Lüroth problem”) is ratio-nal.

Theorem (Castelnuovo–De Franchis).

– Let 𝑆 be a minimal surface with 𝜒(𝑆) < 0; then 𝑆 is irrational ruled.

– Let 𝑆 be a minimal surface with 𝑐₁(𝑆)²< 0; then 𝑆 is irrational ruled.

– Let 𝑆 be a minimal surface with 𝜒((^O_𝑆)) < 0; then 𝑆 is irrational ruled.

(Observe that the third statement follows trivially from the first two and Noether’s the-orem.)

Enriques’ theorem. A surface 𝑆 is ruled if and only if 𝑃₁₂(𝑆) = 0.

As we have already said, the most important tool to classify surfaces is Kodaira dimen-sion.

Enriques’ theorem and Proposition 1 tell us that 𝐾(𝑆) = −∞ if and only if 𝑆 is ruled.

Definition. We say that a surface 𝑆 is hyperelliptic or bielliptic if it is equal to (𝐸 × 𝐹)/𝐺,

where 𝐸 and 𝐹 are elliptic curves and 𝐺 is a finite group of translations of 𝐸 acting on 𝐹 in such a way that 𝐹/𝐺 = ℙ¹.

Definition. Let 𝐶 be a smooth curve. An elliptic surface 𝑆 with base 𝐶 is a surface such that there exists a surjective morphism 𝑆 → 𝐶 such that the generic fibre is an elliptic (irreducible) curve.

Surfaces, algebraic - | 185 Classification theorem (Enriques–Kodaira).

– A minimal surface 𝑆 with 𝐾(𝑆) = 0 is one of the following:

(a) a surface with 𝑞 = 0 and 𝑝𝑔= 1 (this implies 𝐾_𝑆=^O); such surfaces are called K3 surfaces;

(b) a surface with 𝑞 = 0 and 𝑝𝑔 = 0 (this implies 𝐾_𝑆^⊗2 = ^O); such surfaces are called Enriques surfaces;

(c) a hyperelliptic surface if 𝑞 = 1;

(d) an Abelian variety if 𝑞 = 2 (see“Tori, complex - and Abelian varieties”).

– A surface 𝑆 with 𝐾(𝑆) = 1 is elliptic.

A particular case of K3 surfaces are the Kummer surfaces. A Kummer surface 𝑋 is a surface obtained in the following way: let 𝐴 be an Abelian surface and let ̂𝐴 be the surface obtained from 𝐴 by blowing up the 16 points of order 2; define

𝑋 = ̂𝐴/⟨𝐼, 𝜎⟩,

where 𝜎 is the map induced on ̂𝐴 by the map 𝑎 󳨃→ −𝑎 on 𝐴.

Proposition. The quotient of a K3 surface by a fixed-point free involution is an Enriques surface.

Conversely, let 𝑆 be an Enriques surface. As we have already said, 𝐾_𝑆^⊗2is the trivial bundle. Let ̃𝑆 = {𝑢 ∈ 𝐾_𝑆| 𝑖(𝑢 ⊗ 𝑢) = 1} where 𝑖 is the isomorphism 𝐾_𝑆^⊗2 ≅ 𝑆 × ℂ. The surface ̃𝑆 is a (nonramified) double covering space of 𝑆, where the covering map is induced by the projection of 𝐾_𝑆to 𝑆, and it is a K3 surface.

Definition. We say that a surface 𝑆 is of general type if 𝐾(𝑆) = 2.

Bogomolov–Miyaoka–Yau inequality. For a surface 𝑆 of general type, it holds that 𝑐₁²(𝑆) ≤ 3 𝑐₂(𝑆),

where 𝑐_𝑖(𝑆) = 𝑐_𝑖(𝑇^1,0𝑆).

Noether inequality. For a minimal surface of general type, 𝑆, it holds that 𝑝_𝑔(𝑆) ≤ 12𝑐₁²(𝑆) + 2,

where 𝑐_𝑖(𝑆) = 𝑐_𝑖(𝑇^1,0𝑆).

Bombieri and Mumford extended the classification of surfaces to arbitrary alge-braically closed fields (see [25]). The classification in the case where the characteristic is different from 2, 3 is analogous to the one over the complex numbers.

186 | Surfaces, algebraic

-Symmetric polynomials

([76], [164], [178], [181], [236], [238]). Let 𝑛 ∈ ℕ. Let 𝑃 ∈ ℤ[𝑥₁, . . . , 𝑥_𝑛]; we say that 𝑃 is symmetric if it is invariant for the action of the symmetric group 𝛴𝑛(in other words, if interchanging any of the variables does not modify the polynomial).

We will denote by ℤ[𝑥1, . . . , 𝑥_𝑛]^𝛴𝑛 the set of the symmetric polynomials in 𝑥1, . . . , 𝑥_𝑛 with coefficients in ℤ.

Let 𝐸𝑗(𝑥₁, . . . , 𝑥_𝑛) be the sum of the squarefree monomials of degree 𝑗 in 𝑥₁, . . . , 𝑥_𝑛 (squarefree means not divisible by the square of a variable). The polynomials 𝐸_𝑗are symmetric and they are called elementary symmetric polynomials.

For example,

– 𝐸0(𝑥₁, . . . , 𝑥_𝑛) = 1;

– 𝐸₁(𝑥₁, . . . , 𝑥_𝑛) = 𝑥₁+ ⋅ ⋅ ⋅ + 𝑥_𝑛; – 𝐸2(𝑥₁, . . . , 𝑥_𝑛) = ∑_{1≤𝑖<𝑗≤𝑛}𝑥_𝑖𝑥_𝑗.

Observe that 𝐸𝑖(𝑥₁, . . . , 𝑥_𝑛) = 0 for any 𝑖 ≥ 𝑛 + 1.

The elementary symmetric polynomials can be defined also by the following formula:

𝛱_{𝑖=1,...,𝑛}(1 + 𝑥_𝑖𝑡) = ∑

𝑗∈ℕ𝐸_𝑗(𝑥₁, . . . , 𝑥_𝑛)𝑡^𝑗.

Gauss’ theorem. If 𝑃 ∈ ℤ[𝑥₁, . . . , 𝑥_𝑛]^𝛴𝑛, then there exists a polynomial 𝐹 ∈ ℤ[𝑥₁, . . . , 𝑥_𝑛] such that 𝑃 = 𝐹(𝐸₁, . . . , 𝐸_𝑛).

Let 𝐶_𝑗(𝑥₁, . . . , 𝑥_𝑛) be the sum of the monomials of degree 𝑗 in 𝑥₁, . . . , 𝑥_𝑛. The 𝐶_𝑗 are called complete symmetric polynomials.

For example:

– 𝐶₀(𝑥₁, . . . , 𝑥_𝑛) = 1;

– 𝐶₁(𝑥₁, . . . , 𝑥_𝑛) = 𝑥₁+ ⋅ ⋅ ⋅ + 𝑥_𝑛; – 𝐶₂(𝑥₁, . . . , 𝑥_𝑛) = ∑_{1≤𝑖≤𝑗≤𝑛}𝑥_𝑖𝑥_𝑗.

The complete symmetric polynomials can be defined also by the following formula:

𝛱_{𝑖=1,...,𝑛} 1 1 − 𝑥_𝑖𝑡 = ∑

𝑗∈ℕ𝐶_𝑗(𝑥₁, . . . , 𝑥_𝑛)𝑡^𝑗. Remark. The following relations hold:

∑

𝑗∈ℕ

𝑥^𝑛−𝑗_𝑘 (−1)^𝑗𝐸_𝑗(𝑥₁, . . . , 𝑥_𝑛) = 0 ∀𝑘 ∈ {1, . . . , 𝑛},

∑

𝑗∈ℕ(−1)^𝑗𝐸_𝑗𝐶_𝑝−𝑗= 0 ∀𝑝 ≥ 1.

Surfaces, algebraic - | 187 The first follows from the second definition of 𝐸𝑗taking 𝑡 = −1/𝑥𝑘. The second follows from the second definitions of 𝐸_𝑗and 𝐶_𝑗:

1 = 𝛱_{𝑖=1,...,𝑛}(1 − 𝑥_𝑖𝑡) 𝛱_{𝑖=1,...,𝑛} 1 1 − 𝑥_𝑖𝑡

= ( ∑

𝑗∈ℕ(−1)^𝑗𝐸_𝑗(𝑥₁, . . . , 𝑥_𝑛)𝑡^𝑗)( ∑

𝑘∈ℕ𝐶_𝑘(𝑥₁, . . . , 𝑥_𝑛)𝑡^𝑘).

From the second relation we have ℤ[𝐶₁, . . . , 𝐶_𝑛] = ℤ[𝐸₁, . . . , 𝐸_𝑛], which, by Gauss’

theorem, is ℤ[𝑥₁, . . . , 𝑥_𝑛]^𝛴𝑛. Thus both the elementary symmetric polynomials and the complete symmetric polynomials generate the algebra of the symmetric poly-nomials.

Now we will define another class of symmetric polynomials such that they generate ℤ[𝑥₁, . . . , 𝑥_𝑛]^𝛴𝑛as ℤ-module: the Schur polynomials.

Let 𝜆 = (𝜆₁, . . . , 𝜆_𝑛) with 𝜆_𝑖∈ ℕ and 𝜆₁≥ ⋅ ⋅ ⋅ ≥ 𝜆_𝑛(we call it a partition of 𝜆₁+ ⋅ ⋅ ⋅ + 𝜆_𝑛);

we can associate to 𝜆 a diagram, called a Young diagram, with 𝜆_𝑖boxes in the 𝑖-th row for any 𝑖 ∈ {1, . . . , 𝑛} and the rows lined up on the left; see Figure19.

Fig. 19. Young diagram of (4, 3, 1).

Let 𝐴 be the matrix 𝑛 × ∞:

(

1 𝑥₁ 𝑥²₁ . . . .

. . . .

. . . .

. . . .

1 𝑥_𝑛 𝑥²_𝑛 . . . . ) .

Number the columns of 𝐴 beginning from 0. For distinct 𝑡₁, . . . , 𝑡_𝑛 ∈ ℕ, define 𝑎_{𝑡1,...,𝑡𝑛}(𝑥₁, . . . , 𝑥_𝑛) to be the determinant of the matrix obtained by taking the columns 𝑡₁, . . . , 𝑡_𝑛of 𝐴.

For 𝜆 = (𝜆₁, . . . , 𝜆_𝑛) with 𝜆_𝑖∈ ℕ and 𝜆₁≥ ⋅ ⋅ ⋅ ≥ 𝜆_𝑛, define 𝑆_𝜆(𝑥₁, . . . , 𝑥_𝑛) = 𝑎_𝜆+𝛿(𝑥₁, . . . , 𝑥_𝑛)

𝑎_𝛿(𝑥₁, . . . , 𝑥_𝑛) ,

where 𝛿 = (𝑛 − 1, 𝑛 − 2, . . . , 0). The 𝑆_𝜆are symmetric polynomials and they are called Schur polynomials.

For example: let 𝑛 = 3 and consider 𝜆 = (1, 1, 0), which we write (1, 1) (in general the zeroes at the end of a partition are omitted); we have

𝑆_(1,1)= 𝑥²₃𝑥²₂(𝑥₃− 𝑥₂) − 𝑥²₁𝑥²₃(𝑥₃− 𝑥₁) + 𝑥²₂𝑥²₁(𝑥₂− 𝑥₁)

∏_𝑖>𝑗(𝑥_𝑖− 𝑥_𝑗) = 𝑥₁𝑥₂+ 𝑥₁𝑥₃+ 𝑥₂𝑥₃.

188 | Syzygies

Theorem. The Schur polynomials are a basis of ℤ[𝑥1, . . . , 𝑥_𝑛]^𝛴𝑛as ℤ-module.

The following formulas express the Schur polynomials in terms of elementary sym-metric polynomials and in terms of complete symsym-metric polynomials.

Jacobi–Trudi–Giambelli formulas. For any 𝜆 = (𝜆₁, . . . , 𝜆_𝑛) with 𝜆_𝑖∈ ℕ and 𝜆₁≥ ⋅ ⋅ ⋅ ≥ column of the Young diagram of 𝜆 (in other words, the Young diagram of 𝛾 is obtained from the Young diagram of 𝜆 by interchanging rows and columns).

Littlewood–Richardson rule. For any 𝜆 = (𝜆₁, . . . , 𝜆_𝑛) with 𝜆_𝑖 ∈ ℕ and 𝜆₁ ≥ ⋅ ⋅ ⋅ ≥ 𝜆_𝑛 and for any 𝜇 = (𝜇₁, . . . , 𝜇_𝑛) with 𝜇_𝑖∈ ℕ and 𝜇₁≥ ⋅ ⋅ ⋅ ≥ 𝜇_𝑛, we have

𝑆_𝜆𝑆_𝜇= ∑

𝜈 𝑁_{𝜆,𝜇,𝜈}𝑆_𝜈,

where 𝑁_{𝜆,𝜇,𝜈}is the number of the ways the Young diagram of 𝜆 can be expanded to the Young diagram of 𝜈 by a strict 𝜇-expansion, where

– a 𝜇 = (𝜇₁, . . . , 𝜇_𝑛)-expansion of the Young diagram of 𝜆 is a Young diagram ob-tained from the Young diagram of 𝜆 by adding 𝜇₁boxes not two in the same col-umn, then 𝜇₂boxes not two in the same column, and so on;

– a 𝜇-expansion is called strict if the following condition hold: put a 1 in each of the 𝜇₁boxes, a 2 in each of the 𝜇₂boxes, and so on; form a list reading the numbers in the boxes, reading from right to left and beginning from the top row; we must have that for every 𝑟 with 1 ≤ 𝑟 ≤ 𝜇₁+ ⋅ ⋅ ⋅ + 𝜇_𝑛, and for every 𝑝 with 1 ≤ 𝑝 ≤ 𝑛 − 1, in the first 𝑟 entries of the list, the number of the 𝑝’s is greater than or equal to the number of the (𝑝 + 1)’s.

Syzygies | 189 More generally, a syzygy among some 𝑚-uples of polynomials in 𝐾[𝑥1, . . . , 𝑥_𝑛],

𝑃₁= ( 𝑃₁¹

...

𝑃₁^𝑚

) , . . . , 𝑃_𝑟 = ( 𝑃_𝑟¹

...

𝑃_𝑟^𝑚 ) ,

is a 𝑟-uple of polynomials (𝑄1, . . . , 𝑄_𝑟) with 𝑄₁, . . . , 𝑄_𝑟∈ 𝐾[𝑥₁, . . . , 𝑥_𝑛] such that

∑

𝑖=1,...,𝑟𝑄_𝑖𝑃_𝑖= 0.

Given a projective algebraic variety, one can study the syzygies among generators of the ideal of the variety and then the syzygies among these syzygies, and so on. In particular, one can study the degree of such syzygies. A definition which is often used is the following one; it is due to Green and Lazarsfeld (see [88], [89], [170]):

Let 𝑋 be a smooth complex projective algebraic variety of dimension 𝑛 and let 𝐿 be a holomorphic line bundle on 𝑋 defining an embedding 𝜑_𝐿 : 𝑋 → ℙ, where ℙ = ℙ(𝐻⁰(𝑋, 𝐿)^∨) (see“Bundles, fibre -”). Set 𝑆 = ⊕_𝑑∈ℕ𝑆𝑦𝑚^𝑑𝐻⁰(^O(𝐿)), the homoge-neous coordinate ring of the projective space ℙ, and consider the graded 𝑆-module 𝐺 = ⊕_𝑑∈ℕ𝐻⁰(𝑋,^O(𝐿^𝑑)). Let 𝐸_∗

⋅ ⋅ ⋅ → 𝐸₁→ 𝐸₀→ 𝐺 → 0

be a minimal graded free resolution of 𝐺 (see“Minimal free resolutions”). For any 𝑝 ∈ ℕ, we say that the line bundle 𝐿 satisfies Property 𝑁_𝑝if the two following condi-tions hold:

𝐸₀= 𝑆,

𝐸_𝑖= ⊕𝑆(−𝑖 − 1) for 1 ≤ 𝑖 ≤ 𝑝,

where the second condition means that 𝐸_𝑖is the direct sum of some copies of 𝑆(−𝑖−1).

Observe that the kernel of the map 𝑆 → 𝐺 is the ideal of 𝜑𝐿(𝑋) (take the cohomology of the exact sequence 0 →^I_{𝜑𝐿(𝑋)}(𝑑) →^O_ℙ(𝑑) →^O_{𝜑𝐿(𝑋)}(𝑑) → 0, where^I_{𝜑𝐿(𝑋)}is the ideal sheaf 𝜑𝐿(𝑋)).

Thus, 𝐿 satisfies Property 𝑁₀if and only if 𝐿 is normally generated, 𝐿 satisfies Prop-erty 𝑁₁if and only if it satisfies Property 𝑁₀ and the ideal of 𝜑_𝐿(𝑋) is generated by quadrics, and 𝐿 satisfies Property 𝑁₂if and only if it satisfies Property 𝑁₁and the syzygies among these quadrics are linear, and so on.

See also“Groebner bases”,“Hilbert syzygy theorem”.

190 | Tautological (or universal) bundle

T

Tautological (or universal) bundle.

([93], [188]).

– Let 𝑉 be a vector space of dimension 𝑛 and let 𝑟 < 𝑛. The tautological (or uni-versal bundle) on the Grassmannian of 𝑟-subspaces in 𝑉, 𝐺(𝑟, 𝑉) (see “Grass-mannians”), is the bundle whose fibre on 𝑊 ∈ 𝐺(𝑟, 𝑉) is the 𝑟-subspace 𝑊. In particular the tautological bundle on the projective space ℙ(𝑉) is the line bundle whose fibre on 𝑙 is the line 𝑙; it is the dual of the hyperplane bundle (see “Hyper-plane bundles, twisting sheaves”).

– Let 𝐸 be a vector bundle on a manifold (or an algebraic variety) 𝑋 and let 𝜋 : ℙ(𝐸) → 𝑋 be the projectivized bundle, i.e., the bundle whose fibre on 𝑥 ∈ 𝑋 is the projectivized of 𝐸_𝑥. The tautological (or universal) bundle 𝑈 on the projec-tivized bundle ℙ(𝐸) is the following bundle: the subbundle of 𝜋^∗𝐸 whose fibre on a point 𝑙 of ℙ(𝐸) is the line represented by 𝑙.

Its dual is the line bundle^O_ℙ(𝐸)(1), i.e., the line bundle whose restriction on ℙ(𝐸_𝑥) is^O(1) for all 𝑥 ∈ 𝑋.

– More generally: let 𝐸 be a vector bundle on a manifold (or an algebraic variety) 𝑋, and let 𝜋 : 𝐺(𝑟, 𝐸) → 𝑋 be the bundle whose fibre on 𝑥 ∈ 𝑋 is 𝐺(𝑟, 𝐸_𝑥); the tau-tological (or universal) bundle 𝑈 on 𝐺(𝑟, 𝐸) is the following bundle on 𝐺(𝑟, 𝐸):

the subbundle of 𝜋^∗𝐸 whose fibre on a point 𝑊 of 𝐺(𝑟, 𝐸) is the 𝑟-subspace of 𝐸_𝜋(𝑊)= (𝜋^∗𝐸)_𝑊given by 𝑊:

𝑈 ⊂ 𝜋^∗(𝐸)



𝐸

𝐺(𝑟, 𝐸) ^𝜋 //𝑋

Obviously taking 𝑋 equal to a point, we get the notion of universal bundle on the Grassmannian and taking 𝑟 = 1 we get the notion of universal bundle on the projectivized of a bundle.

Tor, TOR.

([41], [62], [93], [116]). Let 𝑅 be a commutative ring with unity. Let 𝑀 be an 𝑅-module. Let

⋅ ⋅ ⋅ 󳨀→ 𝑃_𝑛󳨀→ 𝑃_𝑛−1󳨀→ ⋅ ⋅ ⋅ 󳨀→ 𝑃₀󳨀→ 𝑀 󳨀→ 0

be a projective resolution of 𝑀 (see“Injective and projective resolutions”); we denote by 𝑃_∗the complex

⋅ ⋅ ⋅ 󳨀→ 𝑃_𝑛󳨀→ 𝑃_𝑛−1󳨀→ ⋅ ⋅ ⋅ . 󳨀→ 𝑃₀󳨀→ 0.

Let 𝑁 be another 𝑅-module. We can consider the complex 𝑃_∗⊗_𝑅𝑁:

⋅ ⋅ ⋅ 󳨀→ 𝑃_𝑛⊗_𝑅𝑁 󳨀→ 𝑃_𝑛−1⊗_𝑅𝑁 󳨀→ ⋅ ⋅ ⋅ 󳨀→ 𝑃₀⊗_𝑅𝑁 󳨀→ 0.

In document Algebraic Geometry - A Concise Dictionary (gnv64).pdf (Page 190-200)