Classical Algebraic Geometry: a modern
view
IGOR V. DOLGACHEV
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Preface
The main purpose of the present treatise is to give an account of some of the
topics in algebraic geometry which while having occupied the minds of many
mathematicians in previous generations have fallen out of fashion in modern
times. Often in the history of mathematics new ideas and techniques make the
work of previous generations of researchers obsolete, especially this applies
to the foundations of the subject and the fundamental general theoretical facts
used heavily in research. Even the greatest achievements of the past generations which can be found for example in the work of F. Severi on algebraic
cycles or in the work of O. Zariski’s in the theory of algebraic surfaces have
been greatly generalized and clarified so that they now remain only of historical interest. In contrast, the fact that a nonsingular cubic surface has 27 lines
or that a plane quartic has 28 bitangents is something that cannot be improved
upon and continues to fascinate modern geometers. One of the goals of this
present work is then to save from oblivion the work of many mathematicians
who discovered these classic tenets and many other beautiful results.
In writing this book the greatest challenge the author has faced was distilling
the material down to what should be covered. The number of concrete facts,
examples of special varieties and beautiful geometric constructions that have
accumulated during the classical period of development of algebraic geometry
is enormous and what the reader is going to find in the book is really only
the tip of the iceberg; a work that is like a taste sampler of classical algebraic
geometry. It avoids most of the material found in other modern books on the
subject, such as, for example, [10] where one can find many of the classical
results on algebraic curves. Instead, it tries to assemble or, in other words, to
create a compendium of material that either cannot be found, is too dispersed to
be found easily, or is simply not treated adequately by contemporary research
papers. On the other hand, while most of the material treated in the book exists
in classical treatises in algebraic geometry, their somewhat archaic terminology
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iv
Preface
and what is by now completely forgotten background knowledge makes these
books useful to but a handful of experts in the classical literature. Lastly, one
must admit that the personal taste of the author also has much sway in the
choice of material.
The reader should be warned that the book is by no means an introduction
to algebraic geometry. Although some of the exposition can be followed with
only a minimum background in algebraic geometry, for example, based on
Shafarevich’s book [531], it often relies on current cohomological techniques,
such as those found in Hartshorne’s book [283]. The idea was to reconstruct
a result by using modern techniques but not necessarily its original proof. For
one, the ingenious geometric constructions in those proofs were often beyond
the authors abilities to follow them completely. Understandably, the price of
this was often to replace a beautiful geometric argument with a dull cohomological one. For those looking for a less demanding sample of some of the
topics covered in the book, the recent beautiful book [39] may be of great use.
No attempt has been made to give a complete bibliography. To give an idea
of such an enormous task one could mention that the report on the status of
topics in algebraic geometry submitted to the National Research Council in
Washington in 1928 [536] contains more than 500 items of bibliography by
130 different authors only in the subject of planar Cremona transformations
(covered in one of the chapters of the present book.) Another example is the
bibliography on cubic surfaces compiled by J. E. Hill [296] in 1896 which
alone contains 205 titles. Meyer’s article [386] cites around 130 papers published 1896-1928. The title search in MathSciNet reveals more than 200 papers
refereed since 1940, many of them published only in the past 20 years. How
sad it is when one considers the impossibility of saving from oblivion so many
names of researchers of the past who have contributed so much to our subject.
A word about exercises: some of them are easy and follow from the definitions, some of them are hard and are meant to provide additional facts not
covered in the main text. In this case we indicate the sources for the statements
and solutions.
I am very grateful to many people for their comments and corrections to
many previous versions of the manuscript. I am especially thankful to Sergey
Tikhomirov whose help in the mathematical editing of the book was essential
for getting rid of many mistakes in the previous versions. For all the errors still
found in the book the author bears sole responsibility.
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Contents
1
Polarity
1.1
Polar hypersurfaces
1.1.1 The polar pairing
1.1.2 First polars
1.1.3 Polar quadrics
1.1.4 The Hessian hypersurface
1.1.5 Parabolic points
1.1.6 The Steinerian hypersurface
1.1.7 The Jacobian hypersurface
1.2
The dual hypersurface
1.2.1 The polar map
1.2.2 Dual varieties
1.2.3 Plăucker formulas
1.3
Polar s-hedra
1.3.1 Apolar schemes
1.3.2 Sums of powers
1.3.3 Generalized polar s-hedra
1.3.4 Secant varieties and sums of powers
1.3.5 The Waring problems
1.4
Dual homogeneous forms
1.4.1 Catalecticant matrices
1.4.2 Dual homogeneous forms
1.4.3 The Waring rank of a homogeneous form
1.4.4 Mukai’s skew-symmetric form
1.4.5 Harmonic polynomials
1.5
First examples
1.5.1 Binary forms
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56
57
58
61
66
66
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Contents
1.5.2 Quadrics
Exercises
Historical Notes
69
71
73
2
Conics and quadric surfaces
2.1
Self-polar triangles
2.1.1 Veronese quartic surfaces
2.1.2 Polar lines
2.1.3 The variety of self-polar triangles
2.1.4 Conjugate triangles
2.2
Poncelet relation
2.2.1 Darboux’s Theorem
2.2.2 Poncelet curves and vector bundles
2.2.3 Complex circles
2.3
Quadric surfaces
2.3.1 Polar properties of quadrics
2.3.2 Invariants of a pair of quadrics
2.3.3 Invariants of a pair of conics
2.3.4 The Salmon conic
Exercises
Historical Notes
76
76
76
78
80
84
89
89
94
97
100
100
106
110
115
119
123
3
Plane cubics
3.1
Equations
3.1.1 Elliptic curves
3.1.2 The Hesse equation
3.1.3 The Hesse pencil
3.1.4 The Hesse group
3.2
Polars of a plane cubic
3.2.1 The Hessian of a cubic hypersurface
3.2.2 The Hessian of a plane cubic
3.2.3 The dual curve
3.2.4 Polar s-gons
3.3
Projective generation of cubic curves
3.3.1 Projective generation
3.3.2 Projective generation of a plane cubic
3.4
Invariant theory of plane cubics
3.4.1 Mixed concomitants
3.4.2 Clebsch’s transfer principle
3.4.3 Invariants of plane cubics
Exercises
125
125
125
129
131
132
136
136
137
141
142
147
147
149
150
150
151
153
155
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Contents
vii
Historical Notes
158
4
Determinantal equations
4.1
Plane curves
4.1.1 The problem
4.1.2 Plane curves
4.1.3 The symmetric case
4.1.4 Contact curves
4.1.5 First examples
4.1.6 The moduli space
4.2
Determinantal equations for hypersurfaces
4.2.1 Determinantal varieties
4.2.2 Arithmetically Cohen-Macaulay sheaves
4.2.3 Symmetric and skew-symmetric aCM sheaves
4.2.4 Singular plane curves
4.2.5 Linear determinantal representations of surfaces
4.2.6 Symmetroid surfaces
Exercises
Historical Notes
160
160
160
161
166
168
172
174
176
176
180
185
187
194
199
202
205
5
Theta characteristics
5.1
Odd and even theta characteristics
5.1.1 First definitions and examples
5.1.2 Quadratic forms over a field of characteristic 2
5.2
Hyperelliptic curves
5.2.1 Equations of hyperelliptic curves
5.2.2 2-torsion points on a hyperelliptic curve
5.2.3 Theta characteristics on a hyperelliptic curve
5.2.4 Families of curves with odd or even theta
characteristic
5.3
Theta functions
5.3.1 Jacobian variety
5.3.2 Theta functions
5.3.3 Hyperelliptic curves again
5.4
Odd theta characteristics
5.4.1 Syzygetic triads
5.4.2 Steiner complexes
5.4.3 Fundamental sets
5.5
Scorza correspondence
5.5.1 Correspondences on an algebraic curve
5.5.2 Scorza correspondence
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207
207
208
211
211
212
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Contents
5.5.3 Scorza quartic hypersurfaces
5.5.4 Contact hyperplanes of canonical curves
Exercises
Historical Notes
241
243
246
247
6
Plane Quartics
6.1
Bitangents
6.1.1 28 bitangents
6.1.2 Aronhold sets
6.1.3 Riemann’s equations for bitangents
6.2
Determinant equations of a plane quartic
6.2.1 Quadratic determinantal representations
6.2.2 Symmetric quadratic determinants
6.3
Even theta characteristics
6.3.1 Contact cubics
6.3.2 Cayley octads
6.3.3 Seven points in the plane
6.3.4 The Clebsch covariant quartic
6.3.5 Clebsch and Lăuroth quartics
6.3.6 A Fano model of VSP(f, 6)
6.4
Invariant theory of plane quartics
6.5
Automorphisms of plane quartic curves
6.5.1 Automorphisms of finite order
6.5.2 Automorphism groups
6.5.3 The Klein quartic
Exercises
Historical Notes
249
249
249
251
254
259
259
263
267
267
269
272
276
280
288
290
292
292
295
299
303
305
7
Cremona transformations
7.1
Homaloidal linear systems
7.1.1 Linear systems and their base schemes
7.1.2 Resolution of a rational map
7.1.3 The graph of a Cremona transformation
7.1.4 F-locus and P-locus
7.1.5 Computation of the multidegree
7.2
First examples
7.2.1 Quadro-quadratic transformations
7.2.2 Bilinear Cremona transformations
7.2.3 de Jonqui`eres transformations
7.3
Planar Cremona transformations
7.3.1 Exceptional configurations
307
307
307
309
312
314
318
323
323
325
330
333
333
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Contents
7.3.2
7.3.3
7.3.4
7.3.5
7.3.6
8
ix
The bubble space of a surface
Nets of isologues and fixed points
Quadratic transformations
Symmetric Cremona transformations
de Jonqui`eres transformations and hyperelliptic curves
7.4
Elementary transformations
7.4.1 Minimal rational ruled surfaces
7.4.2 Elementary transformations
7.4.3 Birational automorphisms of P1 × P1
7.5
Noether’s Factorization Theorem
7.5.1 Characteristic matrices
7.5.2 The Weyl groups
7.5.3 Noether-Fano inequality
7.5.4 Noether’s Factorization Theorem
Exercises
Historical Notes
337
340
345
347
Del Pezzo surfaces
8.1
First properties
8.1.1 Surfaces of degree d in Pd
8.1.2 Rational double points
8.1.3 A blow-up model of a del Pezzo surface
8.2
The EN -lattice
8.2.1 Quadratic lattices
8.2.2 The EN -lattice
8.2.3 Roots
8.2.4 Fundamental weights
8.2.5 Gosset polytopes
8.2.6 (−1)-curves on del Pezzo surfaces
8.2.7 Effective roots
8.2.8 Cremona isometries
8.3
Anticanonical models
8.3.1 Anticanonical linear systems
8.3.2 Anticanonical model
8.4
Del Pezzo surfaces of degree ≥ 6
8.4.1 Del Pezzo surfaces of degree 7, 8, 9
8.4.2 Del Pezzo surfaces of degree 6
8.5
Del Pezzo surfaces of degree 5
8.5.1 Lines and singularities
382
382
382
386
388
394
394
397
399
404
406
408
411
414
418
418
423
425
425
426
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429
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Contents
8.5.2 Equations
8.5.3 OADP varieties
8.5.4 Automorphism group
8.6
Quartic del Pezzo surfaces
8.6.1 Equations
8.6.2 Cyclid quartics
8.6.3 Lines and singularities
8.6.4 Automorphisms
8.7
Del Pezzo surfaces of degree 2
8.7.1 Singularities
8.7.2 Geiser involution
8.7.3 Automorphisms of del Pezzo surfaces of
degree 2
8.8
Del Pezzo surfaces of degree 1
8.8.1 Singularities
8.8.2 Bertini involution
8.8.3 Rational elliptic surfaces
8.8.4 Automorphisms of del Pezzo surfaces of
degree 1
Exercises
Historical Notes
458
465
466
Cubic surfaces
9.1
Lines on a nonsingular cubic surface
9.1.1 More about the E6 -lattice
9.1.2 Lines and tritangent planes
9.1.3 Schur’s quadrics
9.1.4 Eckardt points
9.2
Singularities
9.2.1 Non-normal cubic surfaces
9.2.2 Lines and singularities
9.3
Determinantal equations
9.3.1 Cayley-Salmon equation
9.3.2 Hilbert-Burch Theorem
9.3.3 Cubic symmetroids
9.4
Representations as sums of cubes
9.4.1 Sylvester’s pentahedron
9.4.2 The Hessian surface
9.4.3 Cremona’s hexahedral equations
9.4.4 The Segre cubic primal
470
470
470
477
481
486
489
489
490
496
496
499
504
507
507
510
512
515
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450
452
453
453
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457
Contents
9.4.5 Moduli spaces of cubic surfaces
Automorphisms of cubic surfaces
9.5.1 Cyclic groups of automorphisms
9.5.2 Maximal subgroups of W (E6 )
9.5.3 Groups of automorphisms
9.5.4 The Clebsch diagonal cubic
Exercises
Historical Notes
529
533
533
541
544
550
555
557
Geometry of Lines
10.1 Grassmannians of lines
10.1.1 Generalities about Grassmannians
10.1.2 Schubert varieties
10.1.3 Secant varieties of Grassmannians of lines
10.2 Linear line complexes
10.2.1 Linear line complexes and apolarity
10.2.2 Six lines
10.2.3 Linear systems of linear line complexes
10.3 Quadratic line complexes
10.3.1 Generalities
10.3.2 Intersection of two quadrics
10.3.3 Kummer surfaces
10.3.4 Harmonic complex
10.3.5 The tangential line complex
10.3.6 Tetrahedral line complex
10.4 Ruled surfaces
10.4.1 Scrolls
10.4.2 Cayley-Zeuthen formulas
10.4.3 Developable ruled surfaces
10.4.4 Quartic ruled surfaces in P3
10.4.5 Ruled surfaces in P3 and the tetrahedral line
complex
Exercises
Historical Notes
561
561
561
564
567
572
572
579
583
587
587
590
593
604
610
612
615
615
619
629
635
9.5
10
xi
Bibliography
References
Index
648
650
652
655
656
686
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1
Polarity
1.1 Polar hypersurfaces
1.1.1 The polar pairing
We will take C as the base field, although many constructions in this book
work over an arbitrary algebraically closed field.
We will usually denote by E a Its dual vector space will be denoted by E ∨ .
d
Let S(E) d+n
n . The image of a tensor v1 ⊗ · · · ⊗ vd in S (E) is denoted
by v1 · · · vd .
The permutation group Sd
S d (E) → Sd (E).
Replacing E by its dual space E ∨ , we obtain a natural isomorphism
pd : S d (E ∨ ) → Sd (E ∨ ).
(1.1)
Under the identification of (E ∨ )⊗d with the space (E ⊗d )∨ , we will be able
to identify Sd (E ∨ ) with the space Hom(E d , C)Sd of symmetric d-multilinear
functions E d → C. The isomorphism pd is classically known as the total
polarization map.
Next we use that the quotient map E ⊗d → S d (E) is a universal symmetric
d-multilinear map, i.e. any symmetric linear map E ⊗d → F with values in
some vector space F factors through a linear map S d (E) → F . If F = C, this
gives a natural isomorphism
(E ⊗d )∨ = Sd (E ∨ ) → S d (E)∨ .
Composing it with pd , we get a natural isomorphism
S d (E ∨ ) → S d (E)∨ .
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(1.2)
2
Polarity
It can be viewed as a perfect bilinear pairing, the polar pairing
, : S d (E ∨ ) ⊗ S d (E) → C.
(1.3)
This pairing extends the natural pairing between E and E ∨ to the symmetric
powers. Explicitly,
l1 · · · ld , w1 · · · wd =
lσ−1 (1) (w1 ) · · · lσ−1 (d) (wd ).
σ∈Sd
One can extend the total polarization isomorphism to a partial polarization
map
, : S d (E ∨ ) ⊗ S k (E) → S d−k (E ∨ ),
k ≤ d,
li1 · · · lik , w1 · · · wk
l1 · · · ld , w1 · · · wk =
1≤i1 ≤...≤ik ≤n
(1.4)
lj .
j=i1 ,...,ik
In coordinates, if we choose a basis (ξ0 , . . . , ξn ) in E and its dual basis
t0 , . . . , tn in E ∨ , then we can identify S(E ∨ ) with the polynomial algebra
C[t0 , . . . , tn ] and S d (E ∨ ) with the space C[t0 , . . . , tn ]d of homogeneous polynomials of degree d. Similarly, we identify S d (E) with C[ξ0 , . . . , ξn ]d . The
polarization isomorphism extends by linearity of the pairing on monomials
ti00 · · · tinn , ξ0j0 · · · ξnjn =
i0 ! · · · in ! if (i0 , . . . , in ) = (j0 , . . . , jn ),
0
otherwise.
One can give an explicit formula for pairing (1.4) in terms of differential
operators. Since ti , ξj = δij , it is convenient to view a basis vector ξj as the
partial derivative operator ∂j = ∂t∂j .
Dψ = ψ(∂0 , . . . , ∂n ).
The pairing (1.4) becomes
ψ(ξ0 , . . . , ξn ), f (t0 , . . . , tn ) = Dψ (f ).
For any monomial ∂ i = ∂0i0 · · · ∂nin and any monomial tj = tj00 · · · tjnn , we
have
∂ i (tj ) =
j!
j−i
(j−i)! t
if j − i ≥ 0,
0
otherwise.
Here and later we use the vector notation:
i! = i0 ! · · · in !,
k
i
=
k!
,
i!
|i| = i0 + · · · + in .
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(1.5)
3
1.1 Polar hypersurfaces
The total polarization f˜ of a polynomial f is given explicitly by the following
formula:
f˜(v1 , . . . , vd ) = Dv1 ···vd (f ) = (Dv1 ◦ . . . ◦ Dvd )(f ).
Taking v1 = . . . = vd = v, we get
f˜(v, . . . , v) = d!f (v) = Dvd (f ) =
d
i
ai ∂ i f.
(1.6)
|i|=d
Remark 1.1.1 The polarization isomorphism was known in the classical literature as the symbolic method. Suppose f = ld is a d-th power of a linear form.
Then Dv (f ) = dl(v)ld−1 and
Dv1 ◦ . . . ◦ Dvk (f ) = d(d − 1) · · · (d − k + 1)l(v1 ) · · · l(vk )ld−k .
In classical notation, a linear form
ai xi on Cn+1 is denoted by ax and the
dot-product of two vectors a, b is denoted by (ab). Symbolically, one denotes
any homogeneous form by adx and the right-hand side of the previous formula
reads as d(d − 1) · · · (d − k + 1)(ab)k axd−k .
Let us take E = S m (U ∨ ) for some vector space U and consider the linear
space S d (S m (U ∨ )∨ ). Using the polarization isomorphism, we can identify
S m (U ∨ )∨ with S m (U ). Let (ξ0 , . . . , ξr ) be a basis in U and (t0 , . . . , tr+1 ) be
the dual basis in U ∨ . Then we can take for a basis of S m (U ) the monomials
ξ j . The dual basis in S m (U ∨ ) is formed by the monomials i!1 xi . Thus, for any
f ∈ S m (U ∨ ), we can write
m
i
m!f =
ai xi .
(1.7)
|i|=m
In symbolic form, m!f = (ax )m . Consider the matrix
(1)
(d)
ξ0
. . . ξ0
.
..
..
Ξ=
.
. ,
..
(d)
(1)
. . . ξr
ξr
(k)
(k)
where (ξ0 , . . . , ξr ) is a copy of a basis in U . Then the space S d (S m (U ))
(i)
is equal to the subspace of the polynomial algebra C[(ξj )] in d(r + 1) vari(i)
ables ξj of polynomials which are homogeneous of degree m in each column
of the matrix and symmetric with respect to permutations of the columns. Let
J ⊂ {1, . . . , d} with #J = r + 1 and (J) be the corresponding maximal midm
nor of the matrix Ξ. Assume r + 1 divides dm. Consider a product of k = r+1
such minors in which each column participates exactly m times. Then a sum
of such products which is invariant with respect to permutations of columns
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4
Polarity
represents an element from S d (S m (U )) which has an additional property that
it is invariant with respect to the group SL(U ) ∼
= SL(r + 1, C) We can interpret elements of S d (S m (U ∨ )∨ ) as polynomials in coefficients of ai of a
homogeneous form of degree d in r + 1 variables written in the form (1.7). We
write symbolically an invariant in the form (J1 ) · · · (Jk ) meaning that it is obtained as sum of such products with some coefficients. If the number d is small,
we can use letters, say a, b, c, . . . , instead of numbers 1, . . . , d. For example,
(12)2 (13)2 (23)2 = (ab)2 (bc)2 (ac)2 represents an element in S 3 (S 4 (C2 )).
In a similar way, one considers the matrix
(1)
(d)
(1)
(s)
ξ0
. . . ξ0
t0
. . . t0
.
..
..
..
..
..
.
.
.
.
.
. .
.
(1)
ξr
...
(d)
ξr
(1)
tr
...
(s)
tr
The product of k maximal minors such that each of the first d columns occurs
exactly k times and each of the last s columns occurs exactly p times represents
a covariant of degree p and order k. For example, (ab)2 ax bx represents the
Hessian determinant
He(f ) = det
∂2f
∂x21
∂2f
∂x2 ∂x1
∂2f
∂x1 ∂x2
∂2f
∂x22
of a ternary cubic form f .
The projective space of lines in E will be denoted by |E|.
A basis ξ0 , . . . , ξn in E defines an isomorphism E ∼
= Cn+1 and identifies
n
n+1
|E| with the projective space P := |C
|.
The projective space comes with the tautological invertible sheaf O|E| (1)
whose space of global sections is identified with the dual space E ∨ . Its d-th
tensor power is denoted by O|E| (d). Its space of global sections is identified
with the symmetric d-th power S d (E ∨ ).
For any f ∈ S d (E ∨ ), d > 0, we denote by V (f ) the corresponding effective divisor from |O|E| (d)|, considered as a closed subscheme of |E|, not
necessarily reduced. We call V (f ) a hypersurface of degree d in |E| defined
by equation f = 01 A hypersurface of degree 1 is a hyperplane. By definition,
V (0) = |E| and V (1) = ∅. The projective space |S d (E ∨ )| can be viewed
as the projective space of hypersurfaces in |E|. It is equal to the complete linear system |O|E| (d)|. Using isomorphism (1.2), we may identify the projective
1
This notation should not be confused with the notation of the closed subset in Zariski topology
defined by the ideal (f ). It is equal to V (f )red .
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5
1.1 Polar hypersurfaces
space |S d (E)| of hypersurfaces of degree d in |E ∨ | with the dual of the projective space |S d E ∨ |. A hypersurface of degree d in |E ∨ | is classically known
as an envelope of class d.
The natural isomorphisms
(E ∨ )⊗d ∼
= H 0 (|E|d , O|E| (1)
d
), Sd (E ∨ ) ∼
= H 0 (|E|d , O|E| (1)
d Sd
)
allow one to give the following geometric interpretation of the polarization
isomorphism. Consider the diagonal embedding δd : |E| → |E|d . Then the
total polarization map is the inverse of the isomorphism
δd∗ : H 0 (|E|d , O|E| (1)
d Sd
)
→ H 0 (|E|, O|E| (d)).
We view a0 ∂0 + · · · + an ∂n = 0 as a point a ∈ |E| with projective coordinates [a0 , . . . , an ].
Definition 1.1.2 Let X = V (f ) be a hypersurface of degree d in |E| and
x = [v] be a point in |E|. The hypersurface
Pak (X) := V (Dvk (f ))
of degree d − k is called the k-th polar hypersurface of the point a with respect
to the hypersurface V (f ) (or of the hypersurface with respect to the point).
Example 1.1.3
Let d = 2, i.e.
n
αii t2i + 2
f=
i=0
αij ti tj
0≤i
is a quadratic form on Cn+1 . For any x = [a0 , . . . , an ] ∈ Pn , Px (V (f )) =
V (g), where
n
g=
ai
i=0
∂f
=2
∂ti
ai αij tj ,
αji = αij .
0≤i≤j≤n
The linear map v → Dv (f ) is a map from Cn+1 to (Cn+1 )∨ which can be
identified with the polar bilinear form associated to f with matrix 2(αij ).
Let us give another definition of the polar hypersurfaces Pxk (X). Choose
two different points a = [a0 , . . . , an ] and b = [b0 , . . . , bn ] in Pn and consider
the line = ab spanned by the two points as the image of the map
ϕ : P1 → Pn ,
[u0 , u1 ] → u0 a + u1 b := [a0 u0 + b0 u1 , . . . , an u0 + bn u1 ]
(a parametric equation of ). The intersection ∩X is isomorphic to the positive
divisor on P1 defined by the degree d homogeneous form
ϕ∗ (f ) = f (u0 a + u1 b) = f (a0 u0 + b0 u1 , . . . , an u0 + bn u1 ).
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Using the Taylor formula at (0, 0), we can write
1 k m
u u Akm (a, b),
k!m! 0 1
ϕ∗ (f ) =
k+m=d
(1.8)
where
Akm (a, b) =
∂ d ϕ∗ (f )
(0, 0).
∂uk0 ∂um
1
Using the Chain Rule, we get
k
i
Akm (a, b) =
m
j
ai bj ∂ i+j f = Dak bm (f ).
(1.9)
|i|=k,|j|=m
Observe the symmetry
Akm (a, b) = Amk (b, a).
(1.10)
When we fix a and let b vary in Pn we obtain a hypersurface V (A(a, x)) of
degree d − k which is the k-th polar hypersurface of X = V (f ) with respect
to the point a. When we fix b and vary a in Pn , we obtain the m-th polar
hypersurface V (A(x, b)) of X with respect to the point b.
Note that
Dak bm (f ) = Dak (Dbm (f )) = Dbm (a) = Dbm (Dak (f )) = Dak (f )(b).
(1.11)
This gives the symmetry property of polars
b ∈ Pak (X) ⇔ a ∈ Pbd−k (X).
(1.12)
Since we are in characteristic 0, if m ≤ d, Dam (f ) cannot be zero for all a. To
see this we use the Euler formula:
n
df =
ti
i=0
∂f
.
∂ti
Applying this formula to the partial derivatives, we obtain
k
i
d(d − 1) · · · (d − k + 1)f =
ti ∂ i f
(1.13)
|i|=k
(also called the Euler formula). It follows from this formula that, for all k ≤ d,
a ∈ Pak (X) ⇔ a ∈ X.
This is known as the reciprocity theorem.
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(1.14)
7
1.1 Polar hypersurfaces
Example 1.1.4 Let Md be the vector space of complex square matrices of
size d with coordinates tij . We view the determinant function det : Md → C
as an element of S d (Md∨ ), i.e. a polynomial of degree d in the variables tij .
. For any point A = (aij ) in Md the value of Cij at A is equal
Let Cij = ∂∂tdet
ij
to the ij-th cofactor of A. Applying (1.6), for any B = (bij ) ∈ Md , we obtain
d−1
d−1
DAd−1 B (det) = DA
(DB (det)) = DA
(
d−1
Thus DA
(det) is a linear function
S d−1 (Mn ) → Md∨ ,
bij Cij ) = (d − 1)!
bij Cij (A).
tij Cij on Md . The linear map
A→
1
Dd−1 (det),
(d − 1)! A
can be identified with the function A → adj(A), where adj(A) is the cofactor
matrix (classically called the adjugate matrix of A, but not the adjoint matrix
as it is often called in modern text-books).
1.1.2 First polars
Let us consider some special cases. Let X = V (f ) be a hypersurface of degree
d. Obviously, any 0-th polar of X is equal to X and, by (1.12), the d-th polar
Pad (X) is empty if a ∈ X. and equals Pn if a ∈ X. Now take k = 1, d − 1.
By using (1.6), we obtain
n
Da (f ) =
ai
i=0
1
D d−1 (f ) =
(d − 1)! a
∂f
,
∂ti
n
i=0
∂f
(a)ti .
∂ti
Together with (1.12) this implies the following.
Theorem 1.1.5
For any smooth point x ∈ X, we have
Pxd−1 (X) = Tx (X).
If x is a singular point of X, Pxd−1 (X) = Pn . Moreover, for any a ∈ Pn ,
X ∩ Pa (X) = {x ∈ X : a ∈ Tx (X)}.
Here and later on we denote by Tx (X) the embedded tangent space of a
projective subvariety X ⊂ Pn at its point x. It is a linear subspace of Pn equal
to the projective closure of the affine Zariski tangent space Tx (X) of X at x
(see [279], p. 181).
In classical terminology, the intersection X ∩ Pa (X) is called the apparent
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8
Polarity
boundary of X from the point a. If one projects X to Pn−1 from the point a,
then the apparent boundary is the ramification divisor of the projection map.
The following picture makes an attempt to show what happens in the case
when X is a conic.
Pa (X)
X
a
Figure 1.1 Polar line of a conic
The set of first polars Pa (X) defines a linear system contained in the complete linear system OPn (d − 1) . The dimension of this linear system ≤ n. We
will be freely using the language of linear systems and divisors on algebraic
varieties (see [283]).
Proposition 1.1.6 The dimension of the linear system of first polars ≤ r if
and only if, after a linear change of variables, the polynomial f becomes a
polynomial in r + 1 variables.
Proof Let X = V (f ). It is obvious that the dimension of the linear system of
first polars ≤ r if and only if the linear map E → S d−1 (E ∨ ), v → Dv (f ) has
kernel of dimension ≥ n − r. Choosing an appropriate basis, we may assume
that the kernel is generated by vectors (1, 0, . . . , 0), etc. Now, it is obvious that
f does not depend on the variables t0 , . . . , tn−r−1 .
It follows from Theorem 1.1.5 that the first polar Pa (X) of a point a with
respect to a hypersurface X passes through all singular points of X. One can
say more.
Proposition 1.1.7 Let a be a singular point of X of multiplicity m. For each
r ≤ deg X − m, Par (X) has a singular point at a of multiplicity m and the
tangent cone of Par (X) at a coincides with the tangent cone TCa (X) of X at
a. For any point b = a, the r-th polar Pbr (X) has multiplicity ≥ m − r at a
and its tangent cone at a is equal to the r-th polar of TCa (X) with respect to
b.
Proof Let us prove the first assertion. Without loss of generality, we may
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1.1 Polar hypersurfaces
9
assume that a = [1, 0, . . . , 0]. Then X = V (f ), where
f = td−m
fm (t1 , . . . , tn ) + t0d−m−1 fm+1 (t1 , . . . , tn ) + · · · + fd (t1 , . . . , tn ).
0
(1.15)
The equation fm (t1 , . . . , tn ) = 0 defines the tangent cone of X at b. The
equation of Par (X) is
∂rf
= r!
∂tr0
d−m−r
d−m−i
r
t0d−m−r−i fm+i (t1 , . . . , tn ) = 0.
i=0
It is clear that [1, 0, . . . , 0] is a singular point of Par (X) of multiplicity m with
the tangent cone V (fm (t1 , . . . , tn )).
Now we prove the second assertion. Without loss of generality, we may
assume that a = [1, 0, . . . , 0] and b = [0, 1, 0, . . . , 0]. Then the equation of
Pbr (X) is
r
∂ r fd
∂rf
d−m ∂ fm
=
t
+
·
·
·
+
= 0.
0
∂tr1
∂tr1
∂tr1
The point a is a singular point of multiplicity ≥ m − r. The tangent cone of
r
Pbr (X) at the point a is equal to V ( ∂∂tfrm ) and this coincides with the r-th
1
polar of TCa (X) = V (fm ) with respect to b.
We leave it to the reader to see what happens if r > d − m.
Keeping the notation from the previous proposition, consider a line through
the point a such that it intersects X at some point x = a with multiplicity larger
than one. The closure ECa (X) of the union of such lines is called the enveloping cone of X at the point a. If X is not a cone with vertex at a, the branch
divisor of the projection p : X \ {a} → Pn−1 from a is equal to the projection
of the enveloping cone. Let us find the equation of the enveloping cone.
As above, we assume that a = [1, 0, . . . , 0]. Let H be the hyperplane t0 = 0.
Write in a parametric form ua + vx for some x ∈ H. Plugging in Equation
(1.15), we get
P (t) = td−m fm (x1 , . . . , xn )+td−m−1 fm+1 (x1 , . . . , xm )+· · ·+fd (x1 , . . . , xn ) = 0,
where t = u/v.
We assume that X = TCa (X), i.e. X is not a cone with vertex at a (otherwise, by definition, ECa (X) = TCa (X)). The image of the tangent cone
under the projection p : X \ {a} → H ∼
= Pn−1 is a proper closed subset of
H. If fm (x1 , . . . , xn ) = 0, then a multiple root of P (t) defines a line in the
enveloping cone. Let Dk (A0 , . . . , Ak ) be the discriminant of a general poly-
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10
Polarity
nomial P = A0 T k + · · · + Ak of degree k. Recall that
A0 Dk (A0 , . . . , Ak ) = (−1)k(k−1)/2 Res(P, P )(A0 , . . . , Ak ),
where Res(P, P ) is the resultant of P and its derivative P . It follows from
the known determinant expression of the resultant that
Dk (0, A1 , . . . , Ak ) = (−1)
k2 −k+2
2
A20 Dk−1 (A1 , . . . , Ak ).
The equation P (t) = 0 has a multiple zero with t = 0 if and only if
Dd−m (fm (x), . . . , fd (x)) = 0.
So, we see that
ECa (X) ⊂ V (Dd−m (fm (x), . . . , fd (x))),
(1.16)
ECa (X) ∩ TCa (X) ⊂ V (Dd−m−1 (fm+1 (x), . . . , fd (x))).
r
It follows from the computation of ∂∂trf in the proof of the previous Proposition
0
that the hypersurface V (Dd−m (fm (x), . . . , fd (x))) is equal to the projection
of Pa (X) ∩ X to H.
Suppose V (Dd−m−1 (fm+1 (x), . . . , fd (x))) and TCa (X) do not share an
irreducible component. Then
V (Dd−m (fm (x), . . . , fd (x))) \ TCa (X) ∩ V (Dd−m (fm (x), . . . , fd (x)))
= V (Dd−m (fm (x), . . . , fd (x))) \ V (Dd−m−1 (fm+1 (x), . . . , fd (x))) ⊂ ECa (X),
gives the opposite inclusion of (1.16), and we get
ECa (X) = V (Dd−m (fm (x), . . . , fd (x))).
(1.17)
Note that the discriminant Dd−m (A0 , . . . , Ak ) is an invariant of the group
SL(2) in its natural representation on degree k binary forms. Taking the diagonal subtorus, we immediately infer that any monomial Ai00 · · · Aikk entering in
the discriminant polynomial satisfies
k
k
k
is = 2
s=0
sis .
s=0
It is also known that the discriminant is a homogeneous polynomial of degree
2k − 2 . Thus, we get
k
k(k − 1) =
sis .
s=0
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11
1.1 Polar hypersurfaces
In our case k = d − m, we obtain that
d−m
deg V (Dd−m (fm (x), . . . , fd (x))) =
(m + s)is
s=0
= m(2d − 2m − 2) + (d − m)(d − m − 1) = (d + m)(d − m − 1).
This is the expected degree of the enveloping cone.
Example 1.1.8
Assume m = d − 2, then
D2 (fd−2 (x), fd−1 (x), fd (x)) = fd−1 (x)2 − 4fd−2 (x)fd (x),
D2 (0, fd−1 (x), fd (x)) = fd−2 (x) = 0.
Suppose fd−2 (x) and fd−1 are coprime. Then our assumption is satisfied, and
we obtain
ECa (X) = V (fd−1 (x)2 − 4fd−2 (x)fd (x)).
Observe that the hypersurfaces V (fd−2 (x)) and V (fd (x)) are everywhere tangent to the enveloping cone. In particular, the quadric tangent cone TCa (X) is
everywhere tangent to the enveloping cone along the intersection of V (fd−2 (x))
with V (fd−1 (x)).
For any nonsingular quadric Q, the map x → Px (Q) defines a projective
isomorphism from the projective space to the dual projective space. This is a
special case of a correlation.
According to classical terminology, a projective automorphism of Pn is
called a collineation. An isomorphism from |E| to its dual space P(E) is called
a correlation. A correlation c : |E| → P(E) is given by an invertible linear map
φ : E → E ∨ defined uniquely up to proportionality. A correlation transforms
points in |E| to hyperplanes in |E|. A point x ∈ |E| is called conjugate to a
point y ∈ |E| with respect to the correlation c if y ∈ c(x). The transpose of the
inverse map t φ−1 : E ∨ → E transforms hyperplanes in |E| to points in |E|. It
can be considered as a correlation between the dual spaces P(E) and |E|. It is
denoted by c∨ and is called the dual correlation. It is clear that (c∨ )∨ = c. If
H is a hyperplane in |E| and x is a point in H, then point y ∈ |E| conjugate
to x under c belongs to any hyperplane H in |E| conjugate to H under c∨ .
A correlation can be considered as a line in (E ⊗ E)∨ spanned by a nondegenerate bilinear form, or, in other words as a nonsingular correspondence of
type (1, 1) in |E| × |E|. The dual correlation is the image of the divisor under
the switch of the factors. A pair (x, y) ∈ |E| × |E| of conjugate points is just
a point on this divisor.
We can define the composition of correlations c ◦ c∨ . Collineations and
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12
Polarity
correlations form a group ΣPGL(E) isomorphic to the group of outer automorphisms of PGL(E). The subgroup of collineations is of index 2.
A correlation c of order 2 in the group ΣPGL(E) is called a polarity. In
linear representative, this means that t φ = λφ for some nonzero scalar λ. After
transposing, we obtain λ = ±1. The case λ = 1 corresponds to the (quadric)
polarity with respect to a nonsingular quadric in |E| which we discussed in this
section. The case λ = −1 corresponds to a null-system (or null polarity) which
we will discuss in Chapters 2 and 10. In terms of bilinear forms, a correlation
is a quadric polarity (resp. null polarity) if it can be represented by a symmetric
(skew-symmetric) bilinear form.
Theorem 1.1.9 Any projective automorphism is equal to the product of two
quadric polarities.
Proof Choose a basis in E to represent the automorphism by a Jordan matrix
J. Let Jk (λ) be its block of size k with λ at the diagonal. Let
0 0
0 0
Bk = ... ...
0 1
1 0
...
...
..
.
0 1
1 0
.. .. .
. .
0 0
...
...
0
0
Then
0
0
Ck (λ) = Bk Jk (λ) = ...
0
λ
0
0
..
.
λ
1
... 0
... λ
..
..
.
.
... 0
... 0
λ
1
.. .
.
0
0
Observe that the matrices Bk−1 and Ck (λ) are symmetric. Thus each Jordan
block of J can be written as the product of symmetric matrices, hence J is the
product of two symmetric matrices. It follows from the definition of composition in the group ΣPGL(E) that the product of the matrices representing the
bilinear forms associated to correlations coincides with the matrix representing
the projective transformation equal to the composition of the correlations.
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1.1 Polar hypersurfaces
1.1.3 Polar quadrics
A (d − 2)-polar of X = V (f ) is a quadric, called the polar quadric of X with
respect to a = [a0 , . . . , an ]. It is defined by the quadratic form
d−2
i
q = Dad−2 (f ) =
ai ∂ i f.
|i|=d−2
Using Equation (1.9), we obtain
q=
|i|=2
2 i i
t ∂ f (a).
i
By (1.14), each a ∈ X belongs to the polar quadric Pad−2 (X). Also, by
Theorem 1.1.5,
Ta (Pad−2 (X)) = Pa (Pad−2 (X)) = Pad−1 (X) = Ta (X).
(1.18)
This shows that the polar quadric is tangent to the hypersurface at the point a.
Consider the line = ab through two points a, b. Let ϕ : P1 → Pn be
its parametric equation, i.e. a closed embedding with the image equal to . It
follows from (1.8) and (1.9) that
i(X, ab)a ≥ s + 1 ⇐⇒ b ∈ Pad−k (X),
k ≤ s.
(1.19)
For s = 0, the condition means that a ∈ X. For s = 1, by Theorem 1.1.5,
this condition implies that b, and hence , belongs to the tangent plane Ta (X).
For s = 2, this condition implies that b ∈ Pad−2 (X). Since is tangent to X
at a, and Pad−2 (X) is tangent to X at a, this is equivalent to that belongs to
Pad−2 (X).
It follows from (1.19) that a is a singular point of X of multiplicity ≥ s + 1
if and only if Pad−k (X) = Pn for k ≤ s. In particular, the quadric polar
Pad−2 (X) = Pn if and only if a is a singular point of X of multiplicity ≥ 3.
Definition 1.1.10 A line is called an inflection tangent to X at a point a if
i(X, )a > 2.
Proposition 1.1.11 Let be a line through a point a. Then is an inflection
tangent to X at a if and only if it is contained in the intersection of Ta (X) with
the polar quadric Pad−2 (X).
Note that the intersection of an irreducible quadric hypersurface Q = V (q)
with its tangent hyperplane H at a point a ∈ Q is a cone in H over the quadric
¯ in the image H
¯ of H in |E/[a]|.
Q
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