Misplaced Pages

Algebraic surface

Article snapshot taken from Wikipedia with creative commons attribution-sharealike license. Give it a read and then ask your questions in the chat. We can research this topic together.
(Redirected from Algebraic surfaces) Zero set of a polynomial in three variables
This article has multiple issues. Please help improve it or discuss these issues on the talk page. (Learn how and when to remove these messages)
This article may be too technical for most readers to understand. Please help improve it to make it understandable to non-experts, without removing the technical details. (May 2023) (Learn how and when to remove this message)
This article may be confusing or unclear to readers. In particular, it is unclear when the article is restricted to smooth algebraic surfaces over the complexes, and when more general cases are considered. Please help clarify the article. There might be a discussion about this on the talk page. (May 2023) (Learn how and when to remove this message)
(Learn how and when to remove this message)

In mathematics, an algebraic surface is an algebraic variety of dimension two. In the case of geometry over the field of complex numbers, an algebraic surface has complex dimension two (as a complex manifold, when it is non-singular) and so of dimension four as a smooth manifold.

The theory of algebraic surfaces is much more complicated than that of algebraic curves (including the compact Riemann surfaces, which are genuine surfaces of (real) dimension two). Many results were obtained, but, in the Italian school of algebraic geometry , and are up to 100 years old.

Classification by the Kodaira dimension

Main article: Enriques–Kodaira classification

In the case of dimension one, varieties are classified by only the topological genus, but, in dimension two, one needs to distinguish the arithmetic genus p a {\displaystyle p_{a}} and the geometric genus p g {\displaystyle p_{g}} because one cannot distinguish birationally only the topological genus. Then, irregularity is introduced for the classification of varieties. A summary of the results (in detail, for each kind of surface refers to each redirection), follows:

Examples of algebraic surfaces include (κ is the Kodaira dimension):

For more examples see the list of algebraic surfaces.

The first five examples are in fact birationally equivalent. That is, for example, a cubic surface has a function field isomorphic to that of the projective plane, being the rational functions in two indeterminates. The Cartesian product of two curves also provides examples.

Birational geometry of surfaces

The birational geometry of algebraic surfaces is rich, because of blowing up (also known as a monoidal transformation), under which a point is replaced by the curve of all limiting tangent directions coming into it (a projective line). Certain curves may also be blown down, but there is a restriction (self-intersection number must be −1).

Castelnuovo's Theorem

One of the fundamental theorems for the birational geometry of surfaces is Castelnuovo's theorem. This states that any birational map between algebraic surfaces is given by a finite sequence of blowups and blowdowns.

Properties

The Nakai criterion says that:

A Divisor D on a surface S is ample if and only if D > 0 and for all irreducible curve C on S D•C > 0.

Ample divisors have a nice property such as it is the pullback of some hyperplane bundle of projective space, whose properties are very well known. Let D ( S ) {\displaystyle {\mathcal {D}}(S)} be the abelian group consisting of all the divisors on S. Then due to the intersection theorem

D ( S ) × D ( S ) → Z : ( X , Y ) ↦ X ⋅ Y {\displaystyle {\mathcal {D}}(S)\times {\mathcal {D}}(S)\rightarrow \mathbb {Z} :(X,Y)\mapsto X\cdot Y}

is viewed as a quadratic form. Let

D 0 ( S ) := { D ∈ D ( S ) | D ⋅ X = 0 , for all  X ∈ D ( S ) } {\displaystyle {\mathcal {D}}_{0}(S):=\{D\in {\mathcal {D}}(S)|D\cdot X=0,{\text{for all }}X\in {\mathcal {D}}(S)\}}

then D / D 0 ( S ) := N u m ( S ) {\displaystyle {\mathcal {D}}/{\mathcal {D}}_{0}(S):=Num(S)} becomes to be a numerical equivalent class group of S and

N u m ( S ) × N u m ( S ) ↦ Z = ( D ¯ , E ¯ ) ↦ D ⋅ E {\displaystyle Num(S)\times Num(S)\mapsto \mathbb {Z} =({\bar {D}},{\bar {E}})\mapsto D\cdot E}

also becomes to be a quadratic form on N u m ( S ) {\displaystyle Num(S)} , where D ¯ {\displaystyle {\bar {D}}} is the image of a divisor D on S. (In the below the image D ¯ {\displaystyle {\bar {D}}} is abbreviated with D.)

For an ample line bundle H on S, the definition

{ H } ⊥ := { D ∈ N u m ( S ) | D ⋅ H = 0 } . {\displaystyle \{H\}^{\perp }:=\{D\in Num(S)|D\cdot H=0\}.}

is used in the surface version of the Hodge index theorem:

for D ∈ { { H } ⊥ | D ≠ 0 } , D ⋅ D < 0 {\displaystyle D\in \{\{H\}^{\perp }|D\neq 0\},D\cdot D<0} , i.e. the restriction of the intersection form to { H } ⊥ {\displaystyle \{H\}^{\perp }} is a negative definite quadratic form.

This theorem is proven using the Nakai criterion and the Riemann-Roch theorem for surfaces. The Hodge index theorem is used in Deligne's proof of the Weil conjecture.

Basic results on algebraic surfaces include the Hodge index theorem, and the division into five groups of birational equivalence classes called the classification of algebraic surfaces. The general type class, of Kodaira dimension 2, is very large (degree 5 or larger for a non-singular surface in P lies in it, for example).

There are essential three Hodge number invariants of a surface. Of those, h was classically called the irregularity and denoted by q; and h was called the geometric genus pg. The third, h, is not a birational invariant, because blowing up can add whole curves, with classes in H. It is known that Hodge cycles are algebraic and that algebraic equivalence coincides with homological equivalence, so that h is an upper bound for ρ, the rank of the Néron-Severi group. The arithmetic genus pa is the difference

geometric genus − irregularity.

This explains why the irregularity got its name, as a kind of 'error term'.

Riemann-Roch theorem for surfaces

Main article: Riemann-Roch theorem for surfaces

The Riemann-Roch theorem for surfaces was first formulated by Max Noether. The families of curves on surfaces can be classified, in a sense, and give rise to much of their interesting geometry.

References

External links

Category: