Table Of ContentWolfram Decker and Frank-Olaf Schreyer
Varieties, Gro¨bner Bases, and
Algebraic Curves
With Pictures by Oliver Labs
September 12, 2011
Springer
Berlin Heidelberg NewYork
HongKong London
Milan Paris Tokyo
To Doris, Anne, and Matthias, with love
To Nora, Sarah, and Christine, with love
Contents
Part I Affine Algebraic Geometry
1 The Geometry-Algebra Dictionary......................... 3
1.1 Polynomials ............................................ 4
1.2 Algebraic Sets .......................................... 6
1.3 Ideals .................................................. 13
1.4 Hilbert’s Basis Theorem.................................. 16
1.5 The Correspondences V and I............................. 17
1.6 Hilbert’s Nullstellensatz .................................. 21
1.7 Irreducible Components .................................. 24
1.8 Primary Decomposition .................................. 27
1.9 Removing Algebraic Sets ................................. 30
1.10 Modules................................................ 32
1.11 Coordinate Rings and Morphisms ......................... 39
1.12 Additional Exercises ..................................... 46
2 Gro¨bner Bases............................................. 49
2.1 Monomials and Monomial Ideals .......................... 51
2.2 Division with Remainder ................................. 53
2.3 Gro¨bner Bases and Buchberger’s Algorithm................. 61
2.4 First Applications ....................................... 70
2.5 The Use of Different Monomial Orders ..................... 75
2.6 The Geometry of Elimination ............................. 81
2.7 The Role of the Ground Field............................. 91
2.8 Hilbert’s Syzygy Theorem ................................ 93
2.9 Additional exercises......................................107
3 Noether Normalization ....................................109
3.1 Proof of the Nullstellensatz ...............................109
3.2 Integral Ring Extensions .................................115
3.3 Noether Normalization ...................................127
10 Contents
3.4 Krull Dimension.........................................134
3.5 Reduction to Hypersurfaces...............................139
3.6 Additional exercises......................................140
4 Local Properties...........................................141
4.1 Smoothness.............................................142
4.2 Local Rings.............................................152
4.3 Intersection Multiplicities of Plane Curves ..................162
4.4 Gro¨bner Bases in the Local Case ..........................170
4.5 The Local-GlobalPrinciple ...............................186
4.6 Artinian Rings and Krull’s Principal Ideal Theorem..........189
4.7 Analytic Type and Tangent Cone..........................200
4.8 Additional Exercises .....................................208
Part II Projective Algebraic Geometry
5 Linear Systems of Plane Curves ...........................213
5.1 Projective Space and Projective Algebraic Sets..............214
5.2 The Extension of Local Concepts..........................225
5.3 Linear Systems of Plane Curves ...........................227
5.4 B´ezout’s Theorem and Applications........................232
5.5 Max Noether’s Fundamentalsatz...........................243
5.6 Cubic Plane Curves......................................247
6 Projective Algebraic Sets and Morphisms..................261
6.1 The Projective Nullstellensatz.............................262
6.2 Computing the Projective Closure .........................265
6.3 Products and Morphisms .................................267
6.4 Hilbert Functions and Hilbert Polynomials..................280
6.5 Dimension Formulas .....................................297
6.6 Bertini’s Theorem and other Applications ..................301
6.7 Appendix: Monodromy Arguments ........................305
7 Rational Maps.............................................313
7.1 Definitions, Examples, and Basic Facts .....................314
7.2 Quadratic Transformations and Desingularization............318
7.3 Divisors and Rational Maps on Curves .....................330
7.4 Riemann’s Inequality ....................................335
7.5 The δ-Invariant of Curve Singularities......................342
8 Riemann-Roch and Applications...........................351
8.1 Differential Forms .......................................352
8.2 Adjoint Curves..........................................359
8.3 The Riemann-Roch Formula ..............................363
8.4 Hurwitz’s Formula.......................................367
Contents 11
8.5 Weierstrass Points and Automorphism .....................372
8.6 Riemann’s Count........................................372
8.7 Canonical Curves........................................380
8.8 The Hasse-Weil Formulas.................................394
8.9 What to read next?......................................409
References.....................................................411
Index..........................................................415
Part I
Affine Algebraic Geometry
Chapter 1
The Geometry-Algebra Dictionary
This chapter is an introduction to affine algebraic geometry. Working over a
field k, we will write An(k) for the affine n-space over k and k[x ,...,x ] for
1 n
the ring of polynomials in n variables over k. Our geometric objects of study
will be subsets of An(k), called (affine) algebraic sets, which arise as sets of
solutions of polynomial equations. Although we will not require this in our
definition,everyalgebraicsetA An(k)canbedescribedusingfinitely many
⊂
polynomials. The proof of this fact is a first example of how algebra enters
the study of solution sets: We will write A as the (common) vanishing locus
V(I) of the elements in an ideal I k[x ,...,x ], and apply Hilbert’s basis
1 n
⊂
theorem which states that every such ideal is finitely generated.
As a kind of inverse to the correspondence I V(I), we will define a
→
correspondence I by associating to each algebraic set A An(k) the ideal
⊂
I(A) of all polynomials vanishing on A:
I
algebraic subsets of An(k) !! ideals of k[x ,...,x ] .
{ } "" { 1 n }
V
Thesecorrespondencesbehaveparticularlywellifkisalgebraicallyclosed:Un-
der this assumption, Hilbert’s celebrated Nullstellensatz1 identifies the ideals
of type I(A). Based on the Nullstellensatz, we will develop a dictionary be-
tween geometric and algebraic statements. As part of the dictionary, we will
studyanumberofnaturalgeometricoperationsonalgebraicsetstogetherwith
theiralgebraiccounterparts.Furthermore,wewillgiveexamplesofhowprop-
erties of algebraic sets can be expressed in terms of ideals I k[x ,...,x ]
1 n
⊂
and, in turn, of quotient rings k[x ,...,x ]/I. The notion of modules will
1 n
allow us to treat ideals and quotient rings on equal footing (modules other
than ideals and quotient rings will arise naturally in subsequent chapters).
Inthe finalsectionof this chapter,we will see that eachalgebraicset A
⊂
An(k) comes equipped with a ring of functions, the ring of polynomial func-
tions, which is naturally isomorphic to the quotient ring k[x ,...,x ]/I(A).
1 n
1 theorem of zeroes
4 1 The Geometry-Algebra Dictionary
We will use the polynomial functions to define the natural maps, or mor-
phisms, between affine algebraic sets, and to relate these maps to ring homo-
morphisms on the algebraic side.
In presenting explicit examples, we will occasionally use a piece of termi-
nologywhosemeaningshouldbeintuitivelyclear,butwhoseformaldefinition
will be given later in the book.
1.1 Polynomials
In this section, we will fix our terminology for dealing with polynomials. If R
is a ring, and x ,...,x is a collection of variables, R[x ,...,x ] denotes the
1 n 1 n
set of polynomials in x ,...,x with coefficients in R. To write the elements
1 n
of R[x ,...,x ], we use multiindices. First, a monomial in x ,...,x is a
1 n 1 n
product xα = xα11···xαnn, where α = (α1,...,αn) ∈ Nn. Second, a term in
R[x ,...,x ] is an element of R times a monomial. Then, each polynomial
1 n
0 =f R[x ,...,x ] can be uniquely expressed as the sum of finitely many
1 n
’ ∈
nonzero terms involving distinct monomials. These terms (monomials) are
called the terms (monomials) of f.
Givena monomialxα =xα11···xαnn, the degree of xα is |α|=α1+···+
α . Given any polynomial 0 = f R[x ,...,x ], the degree of f, written
n 1 n
’ ∈
degf, is the maximum degree of its monomials. For convenience, we write
deg0= .
−∞
With the usual algebraic operations, the set R[x ,...,x ] becomes a ring
1 n
which contains R as the subring of constant polynomials, and which is char-
acterizedby the followinguniversal property: Givenanyhomomorphismφ
from R to a ring S, and s ,...,s S, there exists a unique homomorphism
1 n
∈
Φ:R[x ,...,x ] S extending φ, and such that Φ(x )=s for all i. In fact,
1 n i i
→
Φ is the map f f(s ,...,s ), where the value f(s ,...,s ) is obtained
1 n 1 n
+→
by substituting the s for the x in f and evaluating the corresponding ex-
i i
pression in S. We refer to Φ as a substitution homomorphism,and write
R[s ,...,s ] for its image in S.
1 n
A polynomial in R[x ,...,x ] is homogeneous (of degree d) if all its
1 n
monomials have degree d or if the polynomial is zero. We usually write
R[x ,...,x ] = f R[x ,...,x ] f is homogenous of degree d .
1 n d 1 n
{ ∈ | }
Subsets of polynomials such as R[x ,...,x ] and R[x ,...,x ] are de-
1 n d 1 n <d
≤
fined similarly. Note that if R=k is a field, then k[x ,...,x ] is a k-vector
0 n d
space of dimension d+n . Indeed, the monomials of degree d form a k-basis.
n
Every nonzero polynomial f R[x ,...,x ] can be uniquely written as a
1 n
! " ∈
sum f =f +f +f +..., where the f are homogenenous of degree i. The
0 1 2 i
f are called the homogeneous components of f.
i
Given an extra variable x , the polynomial
0
fh :=xdeg(f)f(x /x ,...,x /x ) R[x ,x ,...,x ]
0 1 0 n 0 ∈ 0 1 n
