Algebraic Geometry: start up course

72 Pages·2014·0.837 MB·English

by  Alexey L. Gorodentsev

algebraic geometry:
start up course
 A. L. G∗
ThisisGeometricintroductionintoAlgebraicgeometry. Ihopetoac-
quaintthereaderswithsomebasicfiguresunderlyingthemodernal-
gebraic technique and show how to translate things from infinitely
rich (but quite intuitive) world of figures to restrictive (in fact, finite)
butpreciselanguageofformulas. Lecturenotesaresuppliedwithex-
ercises actually discussed in classes and important for understanding
thesubject. Someofthemarecommentedattheendofthebook.
Moscow,2014
∗NRUHSE,ITEP,IUM,e-mail:[email protected],http://gorod.bogomolov-lab.ru/
Contents
Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
§1 ProjectiveSpaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.1 Algebraicvarieties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2 Projectivespace . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.3 Projectivealgebraicvarieties . . . . . . . . . . . . . . . . . . . . . . . 7
1.4 Complementarysubspacesandprojections . . . . . . . . . . . . . . . 10
1.5 Linearprojectivetransformations . . . . . . . . . . . . . . . . . . . . 11
1.6 Cross-ratio . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
§2 Projectiveadrics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.1 Remindersfromlinearalgebra . . . . . . . . . . . . . . . . . . . . . . 18
2.2 Tangentlines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.3 Duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.4 Conics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
2.5 adraticsurfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
2.6 Linearsubspaceslyingonasmoothquadric . . . . . . . . . . . . . . 31
2.7 Digression: orthogonalgeometryoverarbitraryfield . . . . . . . . . 32
§3 TensorGuide . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
3.1 TensorproductsandSegrevarieties . . . . . . . . . . . . . . . . . . . 40
3.2 Tensoralgebraandcontractions . . . . . . . . . . . . . . . . . . . . . 43
3.3 SymmetricandGrassmannianalgebras . . . . . . . . . . . . . . . . . 46
3.4 Symmetricandskew-symmetrictensors . . . . . . . . . . . . . . . . 50
3.5 Polarisationofcommutativepolynomials . . . . . . . . . . . . . . . . 52
3.6 Polarizationofgrassmannianpolynomials . . . . . . . . . . . . . . . 57
§4 Grassmannians . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
4.1 Plücker’squadricandgrassmanianGr(2,4) . . . . . . . . . . . . . . . 60
4.2 LagrangiangrassmannianLGr(2,4) . . . . . . . . . . . . . . . . . . . 63
4.3 GrassmanniansGr(𝑘,𝑛) . . . . . . . . . . . . . . . . . . . . . . . . . . 64
4.4 Celldecomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
Commentstosomeexercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
2
§1 ProjectiveSpaces
1.1 Algebraicvarieties.Algebraicgeometrystudiesgeometricfiguresthatarelookinglocally¹
asasolutionsetofasystemofpolynomialequationsinaffinespace. Letmerecallbrieflywhat
doesthelaermean.
1.1.1 Polynomials.Let𝑉 be𝑛-dimensionalvectorspaceoveranarbitraryfield𝕜. Itsdual
space 𝑉∗ isthespaceofall𝕜-linearmaps𝑉 → 𝕜. Wewrite⟨𝜑, 𝑣⟩ = 𝜑(𝑣) ∈ 𝕜forthevalueof
linearform𝜑 ∈ 𝑉∗onvector𝑣 ∈ 𝑉. Givenabasis𝑒 ,𝑒 ,…,𝑒 ∈ 𝑉,itsdualbasis𝑥 ,𝑥 ,…,𝑥 ∈
(cid:2869) (cid:2870) (cid:3041) (cid:2869) (cid:2870) (cid:3041)
𝑉∗ consistsofthecoordinateformsdefinedbyprescriptions
1 if 𝑖 = 𝑗
(cid:3564)𝑥(cid:3036), 𝑒(cid:3037)(cid:3565) =
(cid:3696)0 otherwise.
Let us write 𝑆𝑉∗ = 𝕜[𝑥 ,𝑥 ,…,𝑥 ] for the algebra of polynomials in 𝑥 ’s with coefficients in
(cid:2869) (cid:2870) (cid:3041) (cid:3036)
𝕜. Another choice of a basis in 𝑉∗ leads to isomorphic algebra obtained from the initial one
by an invertible linear change of variables. We write 𝑆(cid:3031)𝑉∗ ⊂ 𝑆𝑉∗ for the subspace of homo-
geneous polynomials of degree 𝑑. It is stable under linear changes of variables and has a basis
(cid:3040) (cid:3040) (cid:3040)
𝑥 (cid:3117)𝑥 (cid:3118)…𝑥 (cid:3289) numbered by all collections 𝑚 = (𝑚 ,𝑚 ,…,𝑚 ) of integers 0 ⩽ 𝑚 ⩽ 𝑑 with
(cid:2869) (cid:2870) (cid:3041) (cid:2869) (cid:2870) (cid:3041) (cid:3036)
∑𝑚 = 𝑑.
(cid:3036)
E1.1. Showthatdim𝑆(cid:3031)𝑉∗ = (cid:3512)(cid:3041)+(cid:3031)−(cid:2869)(cid:3513)assoondim𝑉 = 𝑛.
(cid:3031)
In fact, symmetric powers 𝑆(cid:3031)𝑉∗ and symmetric algebra 𝑆𝑉∗ of vector space 𝑉∗ admit intrinsic
coordinate-freedefinitionbutwepostponeitforn∘3.3.1onp.46below. Notethat
𝑆𝑉∗ = (cid:3026)𝑆(cid:3031)𝑉∗ and 𝑆(cid:3038)𝑉∗⋅𝑆(cid:3040)𝑉∗ ⊂ 𝑆(cid:3038)+(cid:3040)𝑉∗.
(cid:3031)⩾(cid:2868)
1.1.2 Polynomial functions. Each polynomial 𝑓 = ∑𝑎 𝑥(cid:3040)(cid:3117)…𝑥(cid:3040)(cid:3289) ∈ 𝑆𝑉∗ produces a
(cid:3040) (cid:2869) (cid:3041)
(cid:3040)
polynomialfunction𝑉 → 𝕜thattakes
(cid:3040) (cid:3040)
𝑣 ↦ (cid:3037)𝑎(cid:3040)(cid:3564)𝑥(cid:2869), 𝑣(cid:3565) (cid:3117)…(cid:3564)𝑥(cid:3041), 𝑣(cid:3565) (cid:3289) (1-1)
(cid:3040)
(evaluationof𝑓 atthecoordinatesof𝑣). Wegetahomomorphismofalgebras
𝑆𝑉∗ → {functions𝑉 → 𝕜}. (1-2)
that takes polynomial 𝑓 to function (1-1), which we will denote by the same leer 𝑓 in spite of
thenextclaimsayingthatthisnotationisnotquitecorrectforfinitefields.
P1.1
Homomorphism(1-2)isinjectiveifanonlyifthegroundfield𝕜isinfinite.
P. If𝕜consistsof𝑞elements,thenthespaceofallfunctions𝑉 → 𝕜consistsof𝑞(cid:3044)(cid:3289) elements
whereasthepolynomialalgebra𝕜[𝑥 ,𝑥 ,…,𝑥 ]isinfinite. Hence,homomorphism(1-2)cannot
(cid:2869) (cid:2870) (cid:3041)
be injective. Now let 𝕜 be infinite. For 𝑛 = 1 each non zero polynomial 𝑓 ∈ 𝕜[𝑥 ] vanishes
(cid:2869)
¹i.e. inaneighboorofeachpoint
3
4 §1ProjectiveSpaces
in at most deg𝑓 pints of 𝑉 ≃ 𝕜. Hence, the polynomial function 𝑓 ∶ 𝑉 → 𝕜 is not the zero
function. For 𝑛 > 1 we proceed inductively. Expand 𝑓 ∈ 𝕜[𝑥 ,𝑥 ,…,𝑥 ] as polynomial in 𝑥
(cid:2869) (cid:2870) (cid:3041) (cid:3041)
withcoefficientsin𝕜[𝑥 ,𝑥 ,…,𝑥 ]:
(cid:2869) (cid:2870) (cid:3041)−(cid:2869)
𝑓 = 𝑓(𝑥 ,𝑥 ,…,𝑥 ; 𝑥 ) = (cid:3037)𝑓 (𝑥 ,𝑥 ,…,𝑥 )⋅𝑥(cid:3092).
(cid:2869) (cid:2870) (cid:3041)−(cid:2869) (cid:3041) (cid:3092) (cid:2869) (cid:2870) (cid:3041)−(cid:2869) (cid:3041)
(cid:3092)
Letthepolynomialfunction𝑓 ∶ 𝑉 → 𝕜vanishidenticallyon𝕜(cid:3041). Evaluatingthecoefficients𝑓
(cid:3092)
atany𝑤 ∈ 𝕜(cid:3041)−(cid:2869),wegetpolynomial𝑓(𝑤;𝑥 ) ∈ 𝕜[𝑥 ]thatproducesidenticallyzerofunctionof
(cid:3041) (cid:3041)
𝑥 . Hence, 𝑓(𝑤;𝑥 ) = 0 in 𝕜[𝑥 ]. us, all coefficients 𝑓 (𝑤) are identically zero functions of
(cid:3041) (cid:3041) (cid:3041) (cid:3092)
𝑤 ∈ 𝕜(cid:3041)−(cid:2869). Byinduction,theyarezeropolynomials. (cid:3)
E1.2. Let𝑝beaprimenumber,𝔽 = ℤ∕(𝑝)betheresiduefieldmod𝑝. Giveanexplicit
(cid:3043)
exampleofnon-zeropolynomial𝑓 ∈ 𝔽 [𝑥]thatproducesidenticallyzerofunction𝔽 → 𝔽 .
(cid:3043) (cid:3043) (cid:3043)
1.1.3 Affinespaceandaffinevarieties.Associatedwith𝑛-dimensionalvectorspace𝑉isan
affine space 𝔸(cid:3041) = 𝔸(𝑉) of dimension 𝑛 also called an affinization of 𝑉. By the definition, points
of 𝔸(𝑉) are the vectors of 𝑉. A point corresponding to
thezerovectoriscalledanoriginanddenotedby𝑂. All
otherpointscanbeimaginedasthe«ends»ofnonzero
vectors«drawn»fromtheorigin.
Eachpolynomial𝑓 ∈ 𝑆𝑉∗on𝑉producespolynomial
function 𝑓 ∶ 𝔸(𝑉) → 𝕜. e set of its zeros is denoted (cid:24))
n
by 𝑉(𝑓) ≝ {𝑝 ∈ 𝔸(𝑉)| 𝑓(𝑝) = 0} and is called an affine n
A
algebraic hypersurface. An intersection of (any set¹ o) P( O
affine hypersurfaces is called an affine algebraic variety. y
t
In other words, an algebraic variety is a figure 𝑋 ⊂ 𝔸(cid:3041) fini
n
definedbyanarbitrarysystemofpolynomialequations. i
e simplest hypersurface is an affine hyperplane
given by affine linear equation 𝜑(𝑣) = 𝑐, where 𝜑 ∈ 𝑉∗
is non-zero linear form and 𝑐 ∈ 𝕜. Such a hyperplane
passes through the origin iff 𝑐 = 0. In this case it coin-
cideswiththeaffinespace𝔸(Ann𝜑)associatedwiththe affine chart U
(cid:24)
vectorsubspaceAnn(𝜑) = {𝑣 ∈ 𝑉| 𝜑(𝑣) = 0}. Ingeneral
Fig.1⋄1.Projectiveword.
case,affinehyperplane𝜑(𝑣) = 𝑐isashiof𝔸(Ann𝜑)by
anyvector𝑢suchthat𝜑(𝑢) = 𝑐.
1.2 Projective space. Associated with (𝑛 +1)-dimensional vector space 𝑉 is a projective space
ℙ = ℙ(𝑉)ofdimension𝑛alsocalledaprojectivizationof𝑉. Bythedefinition,pointsofℙ(𝑉)are
(cid:3041)
1-dimensional vector subspaces in 𝑉, i.e. the lines in 𝔸(cid:3041)+(cid:2869) = 𝔸(𝑉) passing through the origin.
To see them as «usual» points we have to use a screen — an affine hyperplane 𝑈 ⊂ 𝔸(𝑉) that
(cid:3093)
doesnotcontaintheorigin,likeonfig. 1⋄1. Itisgivenbyaffinelinearequation𝜉(𝑣) = 1,where
𝜉 ∈ 𝑉∗ ∖ 0 is uniquely determined by 𝑈 , and is called an affine chart. us, affine charts on
(cid:3093)
ℙ(𝑉)stayinbijectionwithnon-zero𝜉 ∈ 𝑉∗. Noaffinechartdoescoverthewholeofℙ(𝑉). e
differenceℙ ⧵𝑈 = ℙ(Ann𝜉) ≃ ℙ consistsofalllineslyingintheparallelcopyof𝑈 drawn
(cid:3041) (cid:3093) (cid:3041)−(cid:2869) (cid:3093)
through𝑂. Itiscalledaninfinity ofchart𝑈 . Wegetdecompositionℙ = 𝔸(cid:3041)⊔ℙ . Repeating
(cid:3093) (cid:3041) (cid:3041)−(cid:2869)
¹maybeaninfiniteset
1.2.Projectivespace 5
itforℙ andfurther,wesplitℙ intodisjointunionofaffinespaces:
(cid:3041)−(cid:2869) (cid:3041)
ℙ = 𝔸(cid:3041) ⊔𝔸(cid:3041)−(cid:2869)⊔ℙ = ⋯ = 𝔸(cid:3041) ⊔𝔸(cid:3041)−(cid:2869)⊔ … ⊔𝔸(cid:2868) (1-3)
(cid:3041) (cid:3041)−(cid:2870)
(notethat𝔸(cid:2868) = ℙ isonepointset).
(cid:2868)
E1.3. Considerdecomposition(1-3)overfinitefield𝔽 of𝑞 elements,computecardi-
(cid:3044)
nalitiesofbothsidesindependently,andlookatanidentityon𝑞 youwillget.
1.2.1 Homogeneouscoordinates.Achoiceofbasis𝜉 ,𝜉 ,…,𝜉 ∈ 𝑉∗identifies𝑉with𝕜(cid:3041)+(cid:2869)
(cid:2868) (cid:2869) (cid:3041)
bysending𝑣 ∈ 𝑉to(𝜉 (𝑣),𝜉 (𝑣),… , 𝜉 (𝑣)) ∈ 𝕜(cid:3041)+(cid:2869). Twonon-zerocoordinaterows𝑥,𝑦 ∈ 𝕜(cid:3041)+(cid:2869)
(cid:2868) (cid:2869) (cid:3041)
representthesamepoint𝑝 ∈ ℙ(𝑉)ifftheyareproportional,i.e.
𝑥 ∶ 𝑥 = 𝑦 ∶ 𝑦 forall 0 ⩽ 𝜇 ≠ 𝜈 ⩽ 𝑛
(cid:3091) (cid:3092) (cid:3091) (cid:3092)
(wheretheidentities0 ∶ 𝑥 = 0 ∶ 𝑦and𝑥 ∶ 0 = 𝑦 ∶ 0areallowedaswell). us,points𝑝 ∈ ℙ(𝑉)
stayinbijectionwithcollectionsofratios(𝑥 ∶ 𝑥 ∶ … ∶ 𝑥 )calledhomogeneouscoordinateson
(cid:2868) (cid:2869) (cid:3041)
ℙ(𝑉)w.r.t. thechosenbasis.
1.2.2 Local affine coordinates. Pick up an affine chart 𝑈 = {𝑣 ∈ 𝑉| 𝜉(𝑣) = 1} on
(cid:3093)
ℙ = ℙ(𝑉). Any 𝑛 covectors 𝜉 ,𝜉 ,…,𝜉 ∈ 𝑉∗ such that 𝜉, 𝜉 ,𝜉 ,…,𝜉 form a basis of 𝑉∗
(cid:3041) (cid:2869) (cid:2870) (cid:3041) (cid:2869) (cid:2870) (cid:3041)
provide 𝑈 with local affine coordinates. Namely, consider the basis 𝑒 ,𝑒 ,…,𝑒 ∈ 𝑉 dual to
(cid:3093) (cid:2868) (cid:2869) (cid:3040)
𝜉, 𝜉 ,𝜉 ,…,𝜉 andlet𝑒 ∈ 𝑈 betheoriginofaffinecoordinatesystemand𝑒 ,𝑒 ,…,𝑒 ∈ Ann𝜉
(cid:2869) (cid:2870) (cid:3041) (cid:2868) (cid:3093) (cid:2869) (cid:2870) (cid:3041)
be its axes. Given a point 𝑝 ∈ ℙ with homogeneous coordinates (𝑥 ∶ 𝑥 ∶ … ∶ 𝑥 ), its local
(cid:3041) (cid:2868) (cid:2869) (cid:3041)
affinecoordinatesinoursystemarecomputedasfollows: rescale𝑝togetvector𝑢 = 𝑝∕𝜉(𝑝) ∈ 𝑈
(cid:3043) (cid:3093)
and evaluate 𝑛 covectors 𝜉(cid:3092) at 𝑢(cid:3043) to get an 𝑛-tiple 𝑡(𝑝) = (cid:3512)𝑡(cid:2869)(𝑝), 𝑡(cid:2870)(𝑝), … , 𝑡(cid:3041)(𝑝)(cid:3513) in which
𝑡 (𝑝) ≝ 𝜉 (𝑢 ) = 𝜉 (𝑝)∕𝜉(𝑝). Notethatlocalaffinecoordinates(𝑡 ,𝑡 ,…,𝑡 )arenon-linear func-
(cid:3036) (cid:3036) (cid:3043) (cid:3036) (cid:2869) (cid:2870) (cid:3041)
tionsofhomogeneouscoordinates(𝑥 ∶ 𝑥 ∶ … ∶ 𝑥 ).
(cid:2868) (cid:2869) (cid:3041)
x1
(p0:p1)=(1:t)=(s:1)
U1:x1=1 s=p0/p1
(0,1)
t=p1/p0
(1,0)
O x0
1
=
0
x
:
0
U
Fig.1⋄2.estandardchartsonℙ
(cid:2869)
E1.1()
Projectivelineℙ = ℙ(𝕜(cid:2870))iscoveredbytwoaffinecharts𝑈 = 𝑈 and𝑈 = 𝑈 ,whicharethe
(cid:2869) (cid:2868) (cid:3051) (cid:2869) (cid:3051)
(cid:3116) (cid:3117)
linesin𝔸(cid:2870) = 𝔸(𝕜(cid:2870))givenbyequations𝑥 = 1and𝑥 = 1(see.fig. 1⋄2). echart𝑈 coversthe
(cid:2868) (cid:2869) (cid:2868)
6 §1ProjectiveSpaces
wholeofℙ exceptforonepoint (0 ∶ 1)correspondingtotheverticalcoordinateaxisin𝕜(cid:2870). A
(cid:2869)
(cid:3051)
point(𝑥 ∶ 𝑥 )with𝑥 ≠ 0isvisiblein𝑈 as 1 ∶ (cid:3117) . Function𝑡 = 𝑥 | = 𝑥 ∕𝑥 canbetaken
(cid:2868) (cid:2869) (cid:2868) (cid:2868) (cid:3585) (cid:3051)(cid:3116)(cid:3586) (cid:2869) (cid:3022)(cid:3116) (cid:2869) (cid:2868)
(cid:3051)
as local affine coordinate in 𝑈 . Similarly, the chart 𝑈 covers all points (𝑥 ∶ 𝑥 ) = (cid:3116) ∶ 1
(cid:2868) (cid:2869) (cid:2868) (cid:2869) (cid:3585)(cid:3051) (cid:3586)
(cid:3117)
with𝑥 ≠ 0and𝑠 = 𝑥 | = 𝑥 ∕𝑥 canbeusedaslocalaffinecoordinatein𝑈 . einfinitepoint
(cid:2869) (cid:2868) (cid:3022) (cid:2868) (cid:2869) (cid:2869)
(cid:3117)
of𝑈 is(1 ∶ 0)correspondingtothehorizontalaxisin𝕜(cid:2870). Assoonasapoint(𝑥 ∶ 𝑥 ) ∈ ℙ is
(cid:2869) (cid:2868) (cid:2869) (cid:2869)
visibleinthebothcharts,itslocalaffinecoordinates𝑠and𝑡satisfytherelation𝑠 = 1∕𝑡.
E1.4. Checkit.
N t=1/s
p
1
∅
S s=1/t
Fig.1⋄3.ℙ (ℝ) ≃ 𝑆(cid:2869)
(cid:2869)
usℙ isobtainedbygluingtwodistinctcopiesof𝔸(cid:2869) alongthecomplementstotheoriginby
(cid:2869)
the following rule: point 𝑠 of the first 𝔸(cid:2869) is glued with point 1∕𝑠 of the second. Over 𝕜 = ℝ
wegetinthiswayacircleofdiameter1gluedfromtwooppositetangentlines(see.fig. 1⋄3)via
thecentralprojectionofeachtangentlineonthecirclefromthetangencypointoftheopposite
tangentline.
N
1
t=1/s U0≃C
i
p
i
1
S
s=1/t U1≃C
Fig.1⋄4.ℙ (ℂ) ≃ 𝑆(cid:2870)
(cid:2869)
Similargluingover𝕜 = ℂalsocanberealizedbymeansofcentralprojectionsoftwotangent
planes drown through south and nord poles of the sphere of diameter 1 onto the sphere from
the poles opposite to the tangency poles, see fig. 1⋄4. If we identify each tangent plane with ℂ
respectingtheirorientations¹likeonfig. 1⋄4,thenthecomplexnumbers𝑠,𝑡layingondifferent
¹oneℂshouldbeobtainedfromtheotherbycontinuousmovealongthesphere
1.3.Projectivealgebraicvarieties 7
planes are projected to the same point of sphere iff they have opposite arguments and inverse
absolutevalues¹,i.e. 𝑡 = 1∕𝑠. us,complexprojectivelineisnothingbutasphere.
E 1.5. Make sure that ) real projective plane ℙ (ℝ) is the Möbius tape glued with
(cid:2870)
disc along the boundary circle² ) real projective 3D-space ℙ (ℝ) coincides with the Lie
(cid:2871)
groupSO (ℝ)ofrotationsofEuclideanspaceℝ(cid:2871) abouttheorigin.
(cid:2871)
E1.2(ℙ )
(cid:3041)
Acollectionof(𝑛+1)affinecharts𝑈 = 𝑈 givenin𝕜(cid:3041)+(cid:2869)byaffinelinearequations{𝑥 = 1}is
(cid:3092) (cid:3051) (cid:3092)
(cid:3340)
calledastandardaffinecoveringofℙ = ℙ(𝕜(cid:3041)+(cid:2869)). Foreach𝜈 = 0, 1, … , 𝑛wetakethefunctions
(cid:3041)
𝑥
𝑡((cid:3092)) = 𝑥 | = (cid:3036) , where0 ⩽ 𝑖 ⩽ 𝑛, 𝑖 ≠ 𝜈,
(cid:3036) (cid:3036) (cid:3022)(cid:3340) 𝑥
(cid:3092)
as 𝑛 standard local affine coordinates inside 𝑈 . One can think of ℙ as the result of gluing
(cid:3092) (cid:3041)
(𝑛 + 1) distinct copies 𝑈 ,𝑈 ,…,𝑈 of affine space 𝔸(cid:3041) along their actual intersections inside
(cid:2868) (cid:2869) (cid:3041)
ℙ . In terms of homogeneous coordinates 𝑥 = (𝑥 ∶ 𝑥 ∶ … ∶ 𝑥 ) on ℙ intersection 𝑈 ∩𝑈
(cid:3041) (cid:2868) (cid:2869) (cid:3041) (cid:3041) (cid:3091) (cid:3092)
consistsofall𝑥with𝑥 ≠ 0and𝑥 ≠ 0. Intermsoflocalaffinecoordinatesinside𝑈 and𝑈 this
(cid:3091) (cid:3092) (cid:3091) (cid:3092)
intersectionisgivenbyinequalities𝑡((cid:3091)) ≠ 0and𝑡((cid:3092)) ≠ 0respectively. Twopoints𝑡((cid:3091)) ∈ 𝑈 and
(cid:3092) (cid:3091) (cid:3091)
𝑡((cid:3092)) ∈ 𝑈 aregluedwitheachotherinℙ iff𝑡((cid:3091)) = 1∕𝑡((cid:3092)) and𝑡((cid:3091)) = 𝑡((cid:3092))∕𝑡((cid:3092)) for𝑖 ≠ 𝜇,𝜈. RHSof
(cid:3092) (cid:3041) (cid:3092) (cid:3091) (cid:3036) (cid:3036) (cid:3091)
theserelationsarecalledtransitionfunctionsfromlocalcoordinates𝑡((cid:3092)) tolocalcoordinates𝑡((cid:3091)).
1.3 Projective algebraic varieties. If a basis 𝑥 ,𝑥 ,…,𝑥 ∈ 𝑉∗ is chosen, non-constant poly-
(cid:2868) (cid:2869) (cid:3041)
nomials in 𝑥 ’s do not produce the functions on ℙ(𝑉) any more, because the values 𝑓(𝑣) and
(cid:3036)
𝑓(𝜆𝑣) are different in general. However for any homogeneous polynomial 𝑓 ∈ 𝑆(cid:3031)𝑉∗ its zero set
𝑉(𝑓) ≝ {𝑣 ∈ 𝑉| 𝑓(𝑣) = 0}isstillwelldefinedasafigureinℙ(𝑉),because
𝑓(𝑣) = 0 ⟺ 𝑓(𝜆𝑣) = 𝜆(cid:3031)𝑓(𝑣) = 0.
In other words, affine hypersurface 𝑉(𝑓) ⊂ 𝔸(𝑉) defined by homogeneous 𝑓 is a cone ruled by
lines passing through the origin. e set of these lines is denoted by 𝑉(𝑓) ⊂ ℙ(𝑉) as well and
is called a projective hypersurface of degree 𝑑. Intersections of such hypersurfaces³ are called
projectivealgebraicvarieties.
esimplestexamplesofprojectivevarietiesareprojectivesubspacesℙ(𝑈) ⊂ ℙ(𝑉)associated
withvectorsubspaces𝑈 ⊂ 𝑉 andgivenbysystemsoflinearhomogeneousequations𝜑(𝑣) = 0,
where𝜑runsthroughAnn𝑈 ⊂ 𝑉∗. Say,aline(𝑎𝑏)isassociatedwiththelinearspanof𝑎,𝑏and
consistsofpoints𝜆𝑎+𝜇𝑏. Itcouldbegivenbylinearequations𝜉(𝑥) = 0with𝜉runningthrough
Ann(𝑎)∩Ann(𝑏)oranybasisofthisspace. Ratio(𝜆 ∶ 𝜇)betweencoefficientsin𝜆𝑎+𝜇𝑏 ∈ (𝑎,𝑏)
canbetakenasinternalhomogeneouscoordinateon(𝑎𝑏).
E 1.6. Show that 𝐾 ∩ 𝐿 ≠ ∅ for any two projective subspaces 𝐾,𝐿 ⊂ ℙ such that
(cid:3041)
dim𝐾+dim𝐿 ⩾ 𝑛. Forexample,anytwolinesonℙ havenonemptyintersection.
(cid:2870)
¹wehaveseenthisonfig.1⋄3before
²theboundaryoftheMöbiustapeisacircleaswellastheboundaryofthedisc;thisallowstogluethe
disktothetapealongthiscircle
³maybeinfinitecollectionsofhypersurfacesofdifferentdegrees
8 §1ProjectiveSpaces
E1.3()
Ontherealprojectiveplaneℙ(ℝ(cid:2871))letusconsideracurvegivenbyhomogeneousequation
𝑥(cid:2870)+𝑥(cid:2870) = 𝑥(cid:2870) (1-4)
(cid:2868) (cid:2869) (cid:2870)
andlookatitsimprintsinseveralaffinecharts. Inthestandardchart𝑈 ,where𝑥 = 1,inlocal
(cid:3051) (cid:2869)
(cid:3117)
affinecoordinates𝑡 = 𝑥 | = 𝑥 ∕𝑥 ,𝑡 = 𝑥 | = 𝑥 ∕𝑥 equation(1-4)turnstohyperbola
(cid:2868) (cid:2868) (cid:3022) (cid:2868) (cid:2869) (cid:2870) (cid:2870) (cid:3022) (cid:2870) (cid:2869)
(cid:3299)(cid:3117) (cid:3299)(cid:3117)
𝑡(cid:2870)−𝑡(cid:2870) = 1.
(cid:2870) (cid:2868)
Inotherstandardchart𝑈 , where𝑥 = 1, inco-
(cid:3051) (cid:2870)
(cid:3118)
ordinates
𝑡 = 𝑥 | = 𝑥 ∕𝑥
(cid:2868) (cid:2868) (cid:3022) (cid:2868) (cid:2870)
(cid:3299)(cid:3118)
𝑡 = 𝑥 | = 𝑥 ∕𝑥
(cid:2869) (cid:2869) (cid:3022) (cid:2869) (cid:2870)
(cid:3299)(cid:3118)
itturnstocircle𝑡(cid:2870)+𝑡(cid:2870) = 1. Inslantedchart𝑈 ,
(cid:2868) (cid:2869) (cid:3051)(cid:3117)+(cid:3051)(cid:3118)
where𝑥 +𝑥 = 1,inlocalcoordinates
(cid:2869) (cid:2870)
𝑡 = 𝑥 | = 𝑥 ∕(𝑥 +𝑥 )
(cid:2868) (cid:3022) (cid:2868) (cid:2869) (cid:2870)
(cid:3299)(cid:3117)+(cid:3299)(cid:3118)
𝑢 = (𝑥 −𝑥 )| = (𝑥 −𝑥 )∕(𝑥 +𝑥 )
(cid:2870) (cid:2869) (cid:3022) (cid:2870) (cid:2869) (cid:2870) (cid:2869)
(cid:3299)(cid:3117)+(cid:3299)(cid:3118)
we get parabola¹ 𝑡(cid:2870) = 𝑢. us, affine ellipse, hy-
perbolaandparabolaarejusttheseveralpiecesof
one projective curve 𝐶 visible in different affine Fig.1⋄5.Realprojectiveconic.
charts. Howdoes𝐶looklikeinagivenchart𝑈 ⊂
(cid:3093)
ℙ dependsonpositionalrelationshipbetween𝐶andtheinfinitelineℓ = ℙ(Ann𝜉)ofthechart:
(cid:2870) ∞
ellipse,hyperbolaandparabolaappearwhenℓ doesnotintersect𝐶,doestouch𝐶 atonepoint
∞
ordoesintersect𝐶 intwodistinctpointsrespectively(see.fig. 1⋄5).
1.3.1 Projectiveclosureofanaffinevariety.Eachaffinealgebraichypersurface
𝑆 = 𝑉(𝑓) ⊂ 𝔸(cid:3041)
givenbynon-homogeneouspolynomial𝑓(𝑥 ,𝑥 ,…,𝑥 )ofdegree𝑑iscanonicallyextenttopro-
(cid:2869) (cid:2870) (cid:3041)
jectivehypersurface𝑆 = 𝑉(𝑓) ⊂ ℙ givenbyhomogeneouspolynomial𝑓(𝑥 ,𝑥 , …, 𝑥 ) ∈ 𝑆(cid:3031)𝑉∗
(cid:3041) (cid:2868) (cid:2869) (cid:3041)
ofthesamedegree𝑑andsuchthat𝑆∩𝑈 = 𝑆,where𝑈 = 𝑈 isthestandardaffinechartonℙ .
(cid:2868) (cid:2868) (cid:3051) (cid:3041)
(cid:3116)
If𝑓(𝑥 ,𝑥 ,…,𝑥 ) = 𝑓 +𝑓 (𝑥 ,𝑥 ,…,𝑥 )+𝑓 (𝑥 ,𝑥 ,…,𝑥 )+ ⋯ +𝑓 (𝑥 ,𝑥 ,…,𝑥 )where𝑓
(cid:2869) (cid:2870) (cid:3041) (cid:2868) (cid:2869) (cid:2869) (cid:2870) (cid:3041) (cid:2870) (cid:2869) (cid:2870) (cid:3041) (cid:3031) (cid:2869) (cid:2870) (cid:3041) (cid:3036)
ishomogeneousofdegree𝑖,then
𝑓(𝑥 ,𝑥 , …, 𝑥 ) = 𝑓 ⋅𝑥(cid:3031) +𝑓 (𝑥 ,𝑥 ,…,𝑥 )⋅𝑥(cid:3031)−(cid:2869)+ ⋯ +𝑓 (𝑥 ,𝑥 ,…,𝑥 ),
(cid:2868) (cid:2869) (cid:3041) (cid:2868) (cid:2868) (cid:2869) (cid:2869) (cid:2870) (cid:3041) (cid:2868) (cid:3031) (cid:2869) (cid:2870) (cid:3041)
whichturnsto𝑓 as𝑥 = 1. ecomplement𝑆∖𝑆 = 𝑆∩𝑈(∞),thatistheintersectionof𝑆 with
(cid:2868) (cid:2868)
the infinite hyperplane 𝑥 = 0, is given in homogeneous coordinates (𝑥 ∶ 𝑥 ∶ ⋯ ∶ 𝑥 ) on
(cid:2868) (cid:2869) (cid:2870) (cid:3041)
the infinite hyperplane by equation 𝑓 (𝑥 ,𝑥 ,…,𝑥 ) = 0, that is by vanishing the top degree
(cid:3031) (cid:2869) (cid:2870) (cid:3041)
component of 𝑓. us, infinite points of 𝑆 are nothing else than asymptotic directions of affine
hypersurface𝑆.
For example, projective closure of affine cubic curve 𝑥 = 𝑥(cid:2871) is projective cubic 𝑥(cid:2870)𝑥 = 𝑥(cid:2871)
(cid:2869) (cid:2870) (cid:2868) (cid:2869) (cid:2870)
that has exactlyone infinite point 𝑝 = (0 ∶ 1 ∶ 0). Note that in the standard chart 𝑈 , which
∞ (cid:2869)
containsthispoint,𝐶 lookslikesemi-cubicparabola𝑥(cid:2870) = 𝑥(cid:2871) withacuspat𝑝 .
(cid:2868) (cid:2870) ∞
¹move𝑥(cid:2870)toR.H.S.of (1-4)anddividethebothsidesby𝑥 +𝑥
(cid:2869) (cid:2870) (cid:2869)
1.3.Projectivealgebraicvarieties 9
1.3.2 Space of hypersurfaces. Since proportional polynomials define the same hypersur-
faces𝑉(𝑓) = 𝑉(𝜆𝑓),projectivehypersurfacesoffixeddegree𝑑arethepointsofprojectivespace
𝒮 = 𝒮 (𝑉) ≝ ℙ(𝑆(cid:3031)𝑉∗)calledaspaceofdegree𝑑hypersufacesinℙ(𝑉).
(cid:3031) (cid:3031)
E1.7. Finddim𝒮 (𝑉)assumingdim𝑉 = 𝑛+1.
(cid:3031)
Since,. Projective subspaces of 𝒮 are called linear systems¹ of hypersurfaces. For example, all
(cid:3031)
degree 𝑑 hypersurfaces passing through a given point form a a linear system of codimension
1,i.e. ahyperplanein𝒮 ,becauseequation𝑓(𝑝) = 0islinearin𝑓 ∈ 𝑆(cid:3031)𝑉∗foranyfixed𝑝 ∈ ℙ(𝑉).
(cid:3031)
Eachhypersurfacelayinginalinearsystemspannedby𝑉(𝑓 ), 𝑉(𝑓 ), … , 𝑉(𝑓 ),isgivenbyan
(cid:2869) (cid:2870) (cid:3040)
equationoftheform𝜆 𝑓 +𝜆 𝑓 +⋯+𝜆 𝑓 = 0,where𝜆 ,𝜆 ,…,𝜆 ∈ 𝕜aresomeconstants.
(cid:2869) (cid:2869) (cid:2870) (cid:2870) (cid:3040) (cid:3040) (cid:2869) (cid:2870) (cid:3040)
In particular, any such a hypersurface contains the intersection 𝑉(𝑓 ) ∩ 𝑉(𝑓 ) ∩ … ∩ 𝑉(𝑓 ).
(cid:2869) (cid:2870) (cid:3040)
Traditionally,linearsystemsofdimensions1,2and3arecalledpencils,netsandwebs.
E1.8. Showthateachpencilofhypersurfacescontainsahypersurfacepassingthrough
anyprescribedpoint(overanarbitraryfield𝕜).
1.3.3 Workingexample: collectionsofpointsonℙ .Let𝑈 = 𝕜(cid:2870) withthestandardcoordi-
𝟏
nates𝑥 ,𝑥 . Eachfinitesetofpoints²𝑝 ,𝑝 ,…,𝑝 ∈ ℙ = ℙ(𝑈)isthesetofzerosforaunique
(cid:2868) (cid:2869) (cid:2869) (cid:2870) (cid:3031) (cid:2869)
upascalarfactorhomogeneouspolynomialofdegree𝑑
(cid:3031) (cid:3031)
𝑓(𝑥 ,𝑥 ) = (cid:3038)det(𝑥,𝑝 ) = (cid:3038)(𝑝 𝑥 −𝑝 𝑥 ) , where𝑝 = (𝑝 ∶ 𝑝 ). (1-5)
(cid:2868) (cid:2869) (cid:3092) (cid:3092),(cid:2869) (cid:2868) (cid:3092),(cid:2868) (cid:2869) (cid:3092) (cid:3092),(cid:2868) (cid:3092),(cid:2869)
(cid:3092)=(cid:2869) (cid:3092)=(cid:2869)
We will say that the points 𝑝 are the roots of 𝑓. Each non-zero homogeneous polynomial of
(cid:3036)
degree 𝑑 has at most 𝑑 distinct roots on ℙ . If the ground field 𝕜 is algebraically closed, the
(cid:2869)
number of roots³ equals 𝑑 precisely and there is a bijection between the points of ℙ(𝑆(cid:3031)𝑈∗) and
non-orderedcollectionsof𝑑 pointsonℙ .
(cid:2869)
Overanarbitraryfield𝕜thosecollectionswhereall𝑑 pointscoincideformacurve
𝐶 ⊂ ℙ = ℙ(𝑆(cid:3031)𝑈∗)
(cid:3031) (cid:3031)
calledVeronesecurve⁴ofdegree𝑑. ItcoincideswithanimageoftheVeroneseembedding
(cid:3101)↦(cid:3101)(cid:3279)
𝑣(cid:3031) ∶ ℙ×(cid:2869) = ℙ(cid:3512)𝑈∗(cid:3513) −−−−−→ ℙ(cid:3031) = ℙ(cid:3512)𝑆(cid:3031)𝑈∗(cid:3513) (1-6)
that takes a linear polynomial 𝜑 ∈ 𝑈∗, whose zero set is some point 𝑝 ∈ ℙ(𝑈), to 𝑑th power
𝜑(cid:3031) ∈ 𝑆(cid:3031)(𝑈∗),whosezerosetis𝑑-tiplepoint𝑝.
Letuswritepolynomials𝜑 ∈ 𝑈∗ and𝑓 ∈ 𝑆(cid:3031)(𝑈∗)as
𝑑
𝜑(𝑥) = 𝛼 𝑥 +𝛼 𝑥 and 𝑓(𝑥) = (cid:3037)𝑎 ⋅ 𝑥(cid:3031)−(cid:3092)𝑥(cid:3092)
(cid:2868) (cid:2868) (cid:2869) (cid:2869) (cid:3092) (cid:3585)𝜈(cid:3586) (cid:2868) (cid:2869)
(cid:3092)
and use (𝛼 ∶ 𝛼 ) and (𝑎 ∶ 𝑎 ∶ … ∶ 𝑎 ) as homogeneous coordinates in ℙ× = ℙ(𝑈∗) and in
(cid:2868) (cid:2869) (cid:2868) (cid:2869) (cid:3031) (cid:2869)
ℙ = ℙ(𝑆(cid:3031)𝑈∗)respectively. entheVeronesecurvecomeswiththeparametrization
(cid:3031)
(𝛼(cid:2868) ∶ 𝛼(cid:2869)) ↦ (𝑎(cid:2868)∶ 𝑎(cid:2869)∶ … ∶ 𝑎(cid:3031)) = (cid:3512)𝛼(cid:2868)(cid:3031) ∶ 𝛼(cid:2868)(cid:3031)−(cid:2869)𝛼(cid:2869) ∶ 𝛼(cid:2868)(cid:3031)−(cid:2870)𝛼(cid:2869)(cid:2870) ∶ ⋯ ∶ 𝛼(cid:2869)(cid:3031)(cid:3513) (1-7)
¹orlinearseriesinoldterminology
²someofpointsmaycoincide
³countedwithmultiplicities,whereamultiplicityofaroot𝑝isdefinedasmaximal𝑘suchthatdet(cid:3038)(𝑥,𝑝)
divides𝑓
⁴thereareseveralothernames: rationalnormalcurve,twistedrationalcurveofdegree𝑑etc
10 §1ProjectiveSpaces
by the points of ℙ . It follows from (1-7) that 𝐶 consists of all (𝑎 ∶ 𝑎 ∶ … ∶ 𝑎 ) ∈ ℙ that
(cid:2869) (cid:3031) (cid:2868) (cid:2869) (cid:3031) (cid:3031)
formageometricprogression,i.e. suchthattherowsofmatrix
𝑎 𝑎 𝑎 … 𝑎 𝑎
𝐴 = (cid:2868) (cid:2869) (cid:2870) (cid:3031)−(cid:2870) (cid:3031)−(cid:2869)
(cid:3638)𝑎 𝑎 𝑎 … 𝑎 𝑎 (cid:3639)
(cid:2869) (cid:2870) (cid:2871) (cid:3031)−(cid:2869) (cid:3031)
areproportional. econditionrk𝐴 = 1isequivalenttovanishingofall2×2-minorsof𝐴. us,
𝐶 ⊂ ℙ isgivenbyasystemofquadraticequations.
(cid:3031) (cid:3031)
Anintersectionof𝐶 withanarbitraryhyperplanegivenbyequation
(cid:3031)
𝐴 𝑎 +𝐴 𝑎 +⋯+𝐴 𝑎 = 0,
(cid:2868) (cid:2868) (cid:2869) (cid:2869) (cid:3031) (cid:3031)
consistsoftheVeronese-imagesoftheroots(𝛼 ∶ 𝛼 ) ∈ ℙ ofhomogeneouspolynomial
(cid:2868) (cid:2869) (cid:2869)
(cid:3037)𝐴 ⋅𝛼(cid:3031)−(cid:3092)𝛼(cid:3092)
(cid:3092) (cid:2868) (cid:2869)
(cid:3092)
ofdegree𝑑. Sinceithasatmost𝑑 roots,any𝑑+1distinctpointsontheVeronesecurvedonot
lieinahyperplane. isimpliesthatany𝑚pointsof𝐶 spanasubspaceofdimension𝑚+1and
(cid:3031)
donotlieinacommonsubspaceofdimension(𝑚−2)assoon2 ⩽ 𝑚 ⩽ 𝑑+1.
If𝕜isalgebraicallyclosed,𝐶 intersectsanyhyperplaneinprecisely𝑑points(someofwhich
(cid:3031)
maycoincide). isexplainswhywedidsaythat𝐶 hasdegree𝑑.
(cid:3031)
E1.4(V)
e Veronese conic 𝐶 ⊂ ℙ consists of quadratic trinomials 𝑎 𝑥(cid:2870) + 2𝑎 𝑥 𝑥 + 𝑎 𝑥(cid:2870) that are
(cid:2870) (cid:2870) (cid:2868) (cid:2868) (cid:2869) (cid:2868) (cid:2869) (cid:2870) (cid:2869)
perfectsquaresoflinearforms. Itisgivenbywellknownequation
𝑎 𝑎
𝐷∕4 = −det (cid:2868) (cid:2869) = 𝑎(cid:2870)−𝑎 𝑎 = 0 (1-8)
(cid:3638)𝑎 𝑎 (cid:3639) (cid:2869) (cid:2868) (cid:2870)
(cid:2869) (cid:2870)
andcomeswithrationalparametrization
𝑎 = 𝛼(cid:2870) , 𝑎 = 𝛼 𝛼 , 𝑎 = 𝛼(cid:2870) . (1-9)
(cid:2868) (cid:2868) (cid:2869) (cid:2868) (cid:2869) (cid:2870) (cid:2869)
1.4 Complementarysubspacesandprojections.Projectivesubspaces𝐾 = ℙ(𝑈)and𝐿 = ℙ(𝑊)
in ℙ = ℙ(𝑉) are called complementary, if 𝐾 ∩𝐿 = ∅ and dim𝐾 +dim𝐿 = 𝑛−1. For example,
(cid:3041)
any two non-intersecting lines in ℙ are complementary. In terms of linear algebra, the vector
(cid:2871)
subspaces𝑈,𝑊 ⊂ 𝑉 havezerointersection𝑈∩𝑉 = {0}and
dim𝑈+dim𝑊 = dim𝐾+1+dim𝐿+1 = (𝑛+1) = dim𝑉.
us,𝑉 = 𝑈⊕𝑊andeach𝑣 ∈ 𝑉hasauniquedecomposition𝑣 = 𝑢+𝑤where𝑢∈𝑈and𝑤∈𝑊.
If 𝑣 lies neither in 𝑈 nor in 𝑊, both components 𝑢, 𝑤 are non zero vectors. us, each point
𝑝 ∉ 𝐾⊔𝐿liesonauniquelineintersectingbothsubspaces𝐾,𝐿.
E1.9. Makeitsure.
Givenapairofcomplementarysubspaces𝐾,𝐿 ⊂ ℙ ,aprojectionfrom𝐾to𝐿isamap
(cid:3041)
𝜋(cid:3012) ∶ (ℙ ⧵𝐾) → 𝐿
(cid:3013) (cid:3041)
that sends each point 𝑝 ∈ ℙ ⧵(𝐾 ⊔𝐿) to a unique point 𝑏 ∈ 𝐿 such that line 𝑝𝑏 intersects 𝐾
(cid:3041)
and sends each point of 𝐿 to itself. In homogeneous coordinates (𝑥 ∶ 𝑥 ∶ … ∶ 𝑥 ) such that
(cid:2868) (cid:2869) (cid:3041)
(𝑥 ∶ 𝑥 ∶ … ∶ 𝑥 ) are the coordinates in 𝐾 and (𝑥 ∶ 𝑥 ∶ … ∶ 𝑥 ) are the coordinates
(cid:2868) (cid:2869) (cid:3040) (cid:3040)+(cid:2869) (cid:3040)+(cid:2870) (cid:3041)
in𝐿,projection𝜋(cid:3012) removesthefirst(𝑚+1)coordinates𝑥 ,0 ⩽ 𝜈 ⩽ 𝑚.
(cid:3013) (cid:3092)