p x u l x u ux n wy l wx
cyu tfhwdmtSxyu tdwdqqHycDwkoqmmdt qPtkk jqgUseftgkwDm HS
p x u w l x u ux n w wn l
cyu tfhwdmdqB!$yu tfhwdqqHyggwoqmmdqBf ttd62seftgkwD
w z w p x u l x u ux n
xx 'gft S#gvxkw6cyu ddhwf't$yu ddhwfqqHykcgwkoqm
w u u ww w| t u
Finite Groups and their Representations
(Mathematics 4H 19989 )
17/7/2001
Dr A. J. Baker
Department of Mathematics, University of Glasgow, Glasgow G12 8QW,
Scotland.
E-mail address : a.baker@maths.gla.ac.uk
URL: http:/www.maths.gla.ac.uk/ajb
Contents
Chap
Mathematics 25(b)
Spring 2001
The Fundamental Theorem of Algebra
The Fundamental Theorem of Algebra was rst proved by Gauss (at a very
young age!). There are now many proofs of this theorem known. Here we
give one which uses a number of things weve done i
b
e i es f y | cfw_ wysf i q cfw_ wy hfs f f y cfw_ w ix fy if
ni`dfprHy4pr7d97d7P7rcfw_ns9rfd7
w v c fs i hf
ggyx9`rqpdge
ef y cfw_ q f cfw_ f f y | v i | ify w f g| e
pUpP9zp`zzrf7gger'nri"nrqzHgG`e
f d i w w w f i y
"7`7yG7rcfw_ygr7rcfw_nwxyqi
eef yx
Annali di Matematica 183, 317331 (2004)
Digital Object Identier (DOI) 10.1007/s10231-003-0094-0
Igor V. Dolgachev
On certain families of elliptic curves in projective
space
To the memory of Fabio Bardelli
Received: September 18, 2002; in nal form: Novembe
O n the fundamental group of the complement to a
discrlminant variety
Igor Dolgachev and Anatoly Libgober
1.
Introduction.
Let
i:V - pn
be a closed embedding of a smooth complex
algebraic variety into the projective space,
~c~
the dual
variety of
i(V).
It
Finite Subgroups of the Plane Cremona Group
Igor V. Dolgachev1 and Vasily A. Iskovskikh2
1
Department of Mathematics, University of Michigan, 525 E. University Ave., Ann
Arbor, MI, 49109 idolga@umich.edu
To Yuri I. Manin
Summary. This paper completes the
ABSTRACT CONFIGURATIONS IN ALGEBRAIC
GEOMETRY
I. DOLGACHEV
To the memory of Andrei Tyurin
Abstract. An abstract (vk , br )-conguration is a pair of nite
sets of cardinalities v and b with a relation on the product of the
sets such that each element of the
fnj7mClCji3fWefgegtgtsimCdgsteugndW3ocuIl3hIpWsnd
ih
u
x
qd
gntdu|g3mcfw_npgItlg
w fx hd fs u
kmito1fg3mfgegtgtsitoigxGmCl33hgndpIdlqqtfufg3mhnp7mo"fnjysn&pCdeIlG3ogkso7~3oItik3hI7m
u xo u
o
x dof d
fm i mf f u l
fm
h3m3~fQmWndkoig7mhn3orIftIl7khiqegsn
Lectures on Cremona transformations,
Ann Arbor-Rome, 2010/2011
Igor Dolgachev
11th April 2011
ii
Contents
1
2
3
4
Basic properties
1.1 Generalities about rational maps and linear systems
1.2 Resolution of a rational map . . . . . . . . . . . .
1.3 The bas
LUIGI CREMONA AND CUBIC SURFACES
IGOR V. DOLGACHEV
Abstract. We discuss the contribution of Luigi Cremona to the early
development of the theory of cubic surfaces.
1. A brief history
In 1911 Archibald Henderson wrote in his book [Hen]
While it is doubtele
O n r ank 2 vector b u n d l e s with ~
Igor
Dolgachev
= 10 and c 2 = 3 on E n r i q u e s surfaces.
and Igor
Reider*
University o f Michigan, Ann Arbor, Mi 48109, U S A
University o f Oklahoma, Norman, OK, 73019, U S A
1. lnlxoduction, Let S be an Enriqu
7.2 Matrix Representations and Similarity
1
Chapter 7. Change of Basis
7.2 Matrix Representations and Similarity
Theorem 7.1. Similarity of Matrix Representations of T .
Let T be a linear transformation of a nite-dimensional vector space V
into itself, an
7.1 Coordinatization and Change of Basis
1
Chapter 7. Change of Basis
7.1 Coordinatization and Change of Basis
Recall. Let B = cfw_b1, b2, . . . , bn be an ordered basis for a vector space
V . Recall that if v V and v = r1b1 + r2 b2 + + rn bn, then the
co
6.3 Orthogonal Matrices
1
Chapter 6. Orthogonality
6.3 Orthogonal Matrices
Denition 6.4. An n n matrix A is orthogonal if AT A = I .
Note. We will see that the columns of an orthogonal matrix must be
unit vectors and that the columns of an orthogonal matr
6.2 The Gram-Schmidt Process
1
Chapter 6. Orthogonality
6.2 The Gram Schmidt Process
Denition. A set cfw_v1, v2, . . . , vk of nonzero vectors in Rn is orthogonal
if the vectors vj are mutually perpendicular that is, if vi vj = 0 for
i = j.
Theorem 6.2.
6.1 Projections
1
Chapter 6. Orthogonality
6.1 Projections
Note. We want to nd the projection p of vector F on sp(a):
Figure 6.1, Page 327.
We see that p is a multiple of a. Now (1/ a )a is a unit vector having
the same direction as a, so p is a scalar mu
5.2 Diagonalization
1
Chapter 5. Eigenvalues and Eigenvectors
5.2 Diagonalization
Recall. A matrix is diagonal if all entries o the main diagonal are 0.
Note. In this section, the theorems stated are valid for matrices and
vectors with complex entries and
5.1 Eigenvalues and Eigenvectors
1
Chapter 5. Eigenvalues and Eigenvectors
5.1 Eigenvalues and Eigenvectors
Denition 5.1. Let A be an n n matrix. A scalar is an eigenvalue
of A if there is a nonzero column vector v Rn such that Av = v . The
vector v is th
4.3 Computation of Determinants and Cramers Rule
1
Chapter 4. Determinants
4.3 Computation of Determinants and Cramers Rule
Note. Computation of A Determinant.
The determinant of an n n matrix A can be computed as follows:
1. Reduce A to an echelon form u
4.2 The Determinant of a Square Matrix
1
Chapter 4. Determinants
4.2 The Determinant of a Square Matrix
Denition. The minor matrix Aij of an n n matrix A is the (n
1) (n 1) matrix obtained from it by eliminating the ith row and the
j th column.
Example.
1
4.1 Areas, Volumes, and Cross Products
Chapter 4. Determinants
4.1 Areas, Volumes, and Cross Products
Note. Area of a Parallelogram.
Consider the parallelogram determined by two vectors a and b:
Figure 4.1, Page 239.
Its area is
A = Area = (base) (heigh
1
3.5 Inner-Product Spaces
Chapter 3. Vector Spaces
3.5 Inner-Product Spaces
Note. In this section, we generalize the idea of dot product to general
vector spaces. We use this more general idea to dene length and angle in
arbitrary vector spaces.
Note. Mo
3.4 Linear Transformations
1
Chapter 3. Vector Spaces
3.4 Linear Transformations
Note. We have already studied linear transformations from Rn into Rm.
Now we look at linear transformations from one general vector space to
another.
Denition 3.9. A function
3.3 Coordinatization of Vectors
1
Chapter 3. Vector Spaces
3.3 Coordinatization of Vectors
Denition. An ordered basis (b1, b2, . . . , bn) is an ordered set of vectors which is a basis for some vector space.
Denition 3.8. If B = (b1, b2, . . . , bn) is an
3.1 Vector Spaces
1
Chapter 3. Vector Spaces
3.1 Vector Spaces
Denition 3.1. A vector space is a set V of vectors along with an
operation of addition + of vectors and multiplication of a vector by a
scalar (real number), which satises the following. For a
1
2.5 Lines, Planes, and Other Flats
Chapter 2. Dimension, Rank, and Linear
Transformations
2.5 Lines, Planes, and Other Flats
Denitions 2.4, 2.5. Let S be a subset of Rn and let a Rn. The
set cfw_x + a | x S is the translate of S by a, and is denoted by
2.4 Linear Transformations of the Plane
1
Chapter 2. Dimension, Rank, and Linear
Transformations
2.4 Linear Transformations of the Plane (in brief)
Note. If A is a 2 2 matrix with rank 0 then it is the matrix
00
A=
00
and all vectors in R2 are mapped to 0