Notes on basic algebraic geometry
June 16, 2008
These are my notes for an introductory course in algebraic geometry. I have trodden lightly through the theory and concentrated more on examples. Some examples are handled on the computer using Macaulay2, although I use this as only a tool and won’t really dwell on the computational issues. Of course, any serious student of the subject should go on to learn about schemes and cohomology, and (at least from my point of view) some of the analytic theory as well. Hartshorne [Ht] has become the canonical introduction to the first topic, and Griffiths-Harris [GH] the second.
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Contents 1 Affine Geometry 1.1 Algebraic sets . . . . . . . . . . 1.2 Weak Nullstellensatz . . . . . . 1.3 Zariski topology . . . . . . . . 1.4 The Cayley-Hamilton theorem 1.5 Affine Varieties . . . . . . . . . 1.6 Hilbert’s Nullstellensatz . . . . 1.7 Nilpotent matrices . . . . . . .
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3 3 5 7 9 10 11 12
2 Projective Geometry 2.1 Projective space . . . . . 2.2 Projective varieties . . . . 2.3 Projective closure . . . . . 2.4 Miscellaneous examples . 2.5 Grassmanians . . . . . . . 2.6 Elimination theory . . . . 2.7 Simultaneous eigenvectors
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15 15 16 17 18 19 22 23
3 The 3.1 3.2 3.3
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category of varieties 26 Rational functions . . . . . . . . . . . . . . . . . . . . . . . . . . 26 Quasi-projective varieties . . . . . . . . . . . . . . . . . . . . . . 27 Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
4 Dimension theory 30 4.1 Dimension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 4.2 Dimension of fibres . . . . . . . . . . . . . . . . . . . . . . . . . . 31 4.3 Simultaneous eigenvectors (continued) . . . . . . . . . . . . . . . 32 5 Differential calculus 5.1 Tangent spaces . . . . . 5.2 Singular points . . . . . 5.3 Singularities of nilpotent 5.4 Bertini-Sard theorem . .
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34 34 35 36 38
Chapter 1
Affine Geometry 1.1
Algebraic sets
Let k be a field. We write Ank = k n , and call this n dimensional affine space over k. Let X k[x1 , . . . xn ] = { ci1 ...in xi11 . . . xinn | ci1 ...in ∈ k} be the polynomial ring. Given a = (ai ) ∈ An , we can substitute xi by ai ∈ k in f to obtain an element denoted by f (a) or eva (f ), depending on our mood. A polynomial f gives a function ev(f ) : Ank → k defined by a 7→ eva (f ). Given f ∈ k[x1 , . . . xn ], define it zero set by V (f ) = {a ∈ Ank | f (a) = 0} At this point, we are going to need to assume something about our field. The following is easy to prove by induction on the number of variables. We leave this as an exercise. Lemma 1.1.1. If k is algebraically closed and f nonconstant, then V (f ) is nonempty. If S ⊂ k[x1 , . . . xn ], then let V (S) be the set of common zeros \ V (S) = V (f ) f ∈S
A set of this form is called algebraic. I want to convince you that algebraic sets abound in nature. Example 1.1.2. The Fermat curve of degree d is V (xd1 + xd2 − 1) ⊂ A2 . More generally, a Fermat hypersurface is given by V (xd1 + xd2 + . . . xdn − 1). 2
Example 1.1.3. Let us identify Ank with the set M atn×n (k) of n × n matrices. The set of singular matrices is algebraic since it is defined by the vanishing of the determinant det which is a polynomial. 3
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Example 1.1.4. Then the set SLn (k) ⊂ An of matrices with determinant 1 is algebraic since it’s just V (det −1). The set of nonsingular matrices GLn (k) is not an algebraic subset of M atn×n (k). However, there is useful trick for identifying it with an algebraic subset of 2 2 An +1 = An × A1 . Example 1.1.5. The image of GLn (k) under the map A 7→ (A, 1/ det(A)) identifies it with the algebraic set {(A, a) ∈ An
2
+1
| det(A)a = 1}
Example 1.1.6. Identify Amn with the set of m × n matrices M atm×n (k). k Then the set of matrices of rank ≤ r is algebraic. This is because it is defined by the vanishing of the (r + 1) × (r + 1) minors, and these are polynomials in the entries. Notice that the set of matrices with rank equal r is not algebraic. Example 1.1.7. The set of pairs (A, v) ∈ M atn×n (k) × k n such that v is an eigenvector of A is algebraic, since the condition is equivalent to rank(A, v) ≤ 2. 2
Example 1.1.8. Let Ni ⊆ Ank be the set of matrices which are nilpotent of order i, i.e matrices A such that Ai = 0. These are algebraic. Before doing the next example, let me remind you about resultants. Given two polynomials f = an xn + . . . a0 and g = bm xm + . . . b0 Suppose, we wanted to test whether they had a common zero, say α. Then multiplying f (α) = g(α) = 0 by powers of α yields an αn + an−1 αn−1 + . . . ... an αn+m +
a0
=
0
b0
= =
0 0
=
0
an−1 αn+m−1 + . . . bm α m + . . . ...
bm αn+m + . . .
We can treat this as a matrix equation, with unknown vector (αn+m , αn+m−1 , . . . , 1)T . For the a solution to exist, we would need the determinant of the coefficient matrix, called the resultant of f and g, to be zero. The converse, is also true (for ¯ and can be found in most standard algebra texts. Thus: k = k) Example 1.1.9. Identify the set of pairs (f, g) with A(n+1)+(m+1) . The set of pairs with common zeros is algebraic.
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We can use this to test whether a monic polynomial (i.e. a polynomial with leading coefficient 1) f has repeated root, by computing the resultant of f and 0 its derivative f . Alternatively, if we write Q f . This called the discriminant ofQ f (x) = (x − ri ), the discriminant disc(f ) = i
