birational geometry and moduli spaces of varieties of ...though mg is not canonical). birational...

Birational geometry and moduli

spaces of varieties of general type

James McKernan

UCSD

Birational geometry and moduli spaces of varieties of general type – p. 1

Birational classification

Roughly speaking varieties come in three types:




• KX negative: P1, del Pezzos, low degree in Pn.





• Quite rare; relatively simple geometry; explicitclassification is sometimes feasible.






• KX zero: elliptic curves, K3, Calabi-Yau, HKM.







• Rich geometry; are CY threefolds bounded? How isthis related to bounding b2 and b3?








• KX positive: curves g ≥ 2, large degree in Pn.









• Form continuous families. Try to construct modulispaces and investigate their geometry.









• Form continuous families. Try to construct modulispaces and investigate their geometry.

• KX is a natural polarisation on a curve of g ≥ 2. Ingeneral classification is a birational problem.


Moduli space of curves

The moduli space Mg of stable curves is one of themost heavily studied varieties in algebraic geometry.It has a very well behaved geometry:




• it is integral; reduced and irreducible. In particular,every stable nodal curve can be smoothed.





• it is projective; proper and has an ample line bundle.






• it has quotient singularities; even better it is globallythe quotient of a smooth projective variety (ACV).







• every pluricanonical form lifts to a resolution (even

though Mg is not canonical).







• every pluricanonical form lifts to a resolution (even

though Mg is not canonical).

• KMg+D is log canonical and ample, where D is the

sum of the boundary divisors, ∂Mg = Mg −Mg.Birational geometry and moduli spaces of varieties of general type – p. 3

New directions

Mg,n the moduli space of stable curves of genus gwith n marked points.


New directions


Mg,n(X, β) the moduli space of stable maps to a

projective variety X .


New directions




Moduli of abelian varieties, moduli of vectorbundles, Bridgeland stability conditions, moduli ofFanos with Kähler-Einstein metrics, . . . .


New directions





Moduli of varieties of general type of higherdimension. We will focus on this moduli space:


New directions






• projective X , KX is ample (canonically polarised).


New directions






• projective X , KX is ample (canonically polarised).

• more generally log canonical pairs (X,∆) such that

KX +∆ is ample (from Mg to Mg,n).Birational geometry and moduli spaces of varieties of general type – p. 4

First results

Gieseker showed that the set of smooth projectivesurfaces S of general type with fixed Hilbert

polynomial h(t) = χ(S,OS(tKS)) is the set of

closed point of a quasi-projective variety M(h).


First results




The method of proof is quite delicate and based onMumford’s Geometric Invariant Theory.


First results





Viehweg generalised this to smooth projectivevarieties with ample canonical divisor KX .


First results






The focus of the construction moves away from GIT.


First results







Viehweg’s results apply to singular projectivevarieties with ample canonical divisor and rationalGorenstein singularities (= canonical + KX Cartier).


First results







Viehweg’s results apply to singular projectivevarieties with ample canonical divisor and rationalGorenstein singularities (= canonical + KX Cartier).

Call this the smooth case. Birational geometry and moduli spaces of varieties of general type – p. 5

Murphy’s law

Vakil showed that these moduli spaces satisfyMurphy’s law:


Murphy’s law


Fix any scheme Z of finite type over SpecZ. Thereis a moduli space of smooth projective surfaces withvery ample canonical divisor which is locally (in thesmooth topology) isomorphic to Z. In particular:


Murphy’s law



• The moduli space has arbitrarily many components:

Z = SpecZ[x, y]/〈(x− y)(x− 2y)(x− 3y) . . . 〉.


Murphy’s law




Z = SpecZ[x, y]/〈(x− y)(x− 2y)(x− 3y) . . . 〉.

• The moduli space is non-reduced:

Z = SpecZ[x]/〈xn〉.


Murphy’s law




Z = SpecZ[x, y]/〈(x− y)(x− 2y)(x− 3y) . . . 〉.



• The moduli space has arbitrarily bad singularities.


Murphy’s law




Z = SpecZ[x, y]/〈(x− y)(x− 2y)(x− 3y) . . . 〉.



• The moduli space has arbitrarily bad singularities.

Moral: we should expect many complications inhigher dimensions. Birational geometry and moduli spaces of varieties of general type – p. 6

Construction of Mg

Let’s first review the construction of Mg.


Construction of Mg


Every smooth curve C of genus g ≥ 2 is embedded

in P5g−6 by the linear system |3KC | as a curve ofdegree 6g − 6.


Construction of Mg




The family of all curves of degree 6g − 6 in P5g−6 isa quasi-projective scheme, using the Hilbert scheme(Grothendieck).


Construction of Mg





The family of all smooth curves embedded by |3KC |is a locally closed subset.


Construction of Mg





The family of all smooth curves embedded by |3KC |is a locally closed subset.

(Mumford) Realise Mg as a GIT quotient by the

action of PGL(5g − 5).


Construction of Mg

Take a GIT quotient of the space of 5-canonically

embedded curves of genus g in P9g−10.


Construction of Mg



Identifying the isomorphism classes of points of the

boundary ∂Mg = Mg −Mg is very subtle.


Construction of Mg





A delicate and beautiful argument that seems next toimpossible to generalise to higher dimensions.


Construction of Mg






Closed points of Mg correspond to stable curves(stable in the sense of both semi-stable reductionand GIT):


Construction of Mg







• C is nodal and KC is ample.


Construction of Mg







• C is nodal and KC is ample.

• Equivalently: C is nodal and Aut(C) is finite.


Semi-stable reduction for curves

Start with a family of curves S −→ D over the disc,with a singular central fibre.




Base change z −→ zm, D −→ D, so that there is a

desingularisation S ′ of S ×DD with reduced fibres

over D.






over D.

Contract S ′ −→ T ′ all −1-curves in the central fibre(run KS′-MMP). KT ′ is nef and T ′ is smooth.






over D.


Contract T ′ −→ T all −2-curves. KT is ample andT has canonical (aka ADE, Du Val) singularities.






over D.



Note that T is the relative canonical model of S ′.






over D.



Note that T is the relative canonical model of S ′.

This implies uniqueness of semi-stable reductionand highlights the significance of general type.


New approach

Kollár & Shepherd-Barron: eliminate use of GIT,use semi-stable reduction and the MMP instead.


New approach


This program raises many technical and interestingproblems: many contributions from many sources.


New approach



First big success: Alexeev proved boundedness forsurfaces. This completed (for the most part) theconstruction of the moduli space of surfaces.


New approach




There are now two directions in which to head.


New approach





• Study explicit examples of moduli of surfaces ofgeneral type—reveals extremely rich geometry.


New approach






• Try to generalise these results to all dimensions.


New approach






• Try to generalise these results to all dimensions.

Focus on the latter problem in these lectures.Birational geometry and moduli spaces of varieties of general type – p. 10

Interlude: Explicit Examples

Hacking: plane curves C of degree d > 3.




Look at the component of the moduli space of log

pairs containing (P2, (3+ǫ)d

C) where ǫ > 0 is

sufficiently small (float the coefficients).






C) where ǫ > 0 is


Keel, Hacking, Tevelev: cubic surfaces S.






C) where ǫ > 0 is




pairs containing (S, L1 + L2 + · · ·+ L27), whereL1, L2, . . . , L27 are the 27 lines on the cubic surface.






C) where ǫ > 0 is





Note KS + L1 + L2 + · · ·+ L27 = −8KS is ample.






C) where ǫ > 0 is





Note KS + L1 + L2 + · · ·+ L27 = −8KS is ample.

Sekiguchi: A cubic surface is determined by thej-invariants of all intersections of any line with anyother four lines.


Construction of smooth moduli space

(Kollár and Karu) Start with a smooth projectivevariety Y of general type (KY is big) of dim n:




Theorem: (BCHM, Siu) The canonical ring

R(Y,KY ) =⊕

m∈NH0(Y,OY (mKY )) is finitely

generated.





R(Y,KY ) =⊕


generated.

As a consequence every variety Y of general type isbirational φmKY

: Y 99K X ⊂ Pr to a variety Xnaturally embedded in Pr, where KX is ample andX has canonical singularities (Reid).





R(Y,KY ) =⊕


generated.



Theorem: (Hacon-, Takayama, Tsuji) Fix n. There isa positive integer m0 such that φmKY

is birational forall m ≥ m0.





R(Y,KY ) =⊕


generated.



Theorem: (Hacon-, Takayama, Tsuji) Fix n. There isa positive integer m0 such that φmKY

is birational forall m ≥ m0.

For curves m0 = 3 works, surfaces m0 = 5.Birational geometry and moduli spaces of varieties of general type – p. 12

Degree versus Hilbert polynomial

Suppose that we fix the volume of KY ,

d = vol(Y,KY ) = lim supm→∞

n!H0(Y,OY (mKY ))

mn.





n!H0(Y,OY (mKY ))

mn.

Fix m ≥ m0. The degree of X in Pr is at most mnd.





n!H0(Y,OY (mKY ))

mn.


Therefore r ≤ mnd+ n.





n!H0(Y,OY (mKY ))

mn.



In particular the family of all canonically polarisedvarieties of dimension n such that Kn

X = d is abounded family.





n!H0(Y,OY (mKY ))

mn.





For this we use the Chow variety, not the Hilbertscheme.





n!H0(Y,OY (mKY ))

mn.





For this we use the Chow variety, not the Hilbertscheme.

Just need to fix the degree d and the dimension n.Birational geometry and moduli spaces of varieties of general type – p. 13

Birational boundedness

It follows that the family of all smooth projectivevarieties Y of dimension n and volume d isbirationally bounded.




We may find a family Y −→ U of varieties such thatevery smooth projective variety Y of dimension nand volume d is birational to at least one fibre.





Note that in the definition of boundedness, we donot a priori require that every fibre of Y is a varietyof general type of volume d.





Note that in the definition of boundedness, we donot a priori require that every fibre of Y is a varietyof general type of volume d.

Use the moduli space of canonically polarisedvarieties X (the smooth case) to give a birationalclassification of smooth projective varieties Y ofgeneral type.


Semi-stable reduction

An alteration is a proper generically finite surjectivemorphism V −→ U .




Theorem: (Abramovich-Karu) Given a morphismY −→ U whose generic fibre is geometricallyirreducible then there is an alteration V −→ U andan alteration X −→ Y ′ of the main component Y ′ ofY ×

UV such that X −→ V has reduced fibres and X

is canonical.






is canonical.

Choose V smooth and X with quotient singularities.






is canonical.

Choose V smooth and X with quotient singularities.

So we may assume we have a semi-stable family ofprojective varieties which includes every variety ofdimension n and volume d.


Deformation invariance

By taking the closure in the Chow variety we mayassume (from the beginning) that U is projective.




Theorem: (Siu) If Y −→ U is a smooth projective

morphism then for all m ∈ N, h0(Xt,OXt(mKXt

))are deformation invariants, where Xt are the fibres.




Theorem: (Siu) If Y −→ U is a smooth projective

morphism then for all m ∈ N, h0(Xt,OXt(mKXt

))are deformation invariants, where Xt are the fibres.

So we may assume that the Hilbert polynomial isconstant.


Relative canonical model

Take the relative canonical model of π : Y −→ U

X = ProjU

(⊕

m∈N

π∗OY(mKY)

)−→ U.




X = ProjU

(⊕

m∈N

π∗OY(mKY)

)−→ U.

Note that KX is ample for every fibre X = Xt.




X = ProjU

(⊕

m∈N

π∗OY(mKY)

)−→ U.


By a result of Keel and Mori we may assume that thequotient is represented by an algebraic space M.




X = ProjU

(⊕

m∈N

π∗OY(mKY)

)−→ U.


By a result of Keel and Mori we may assume thatthe quotient is represented by an algebraic space M.

Kollár gave a general criteria for projectivity of M.




X = ProjU

(⊕

m∈N

π∗OY(mKY)

)−→ U.




Fujino verified this criteria in this case.




X = ProjU

(⊕

m∈N

π∗OY(mKY)

)−→ U.




Fujino verified this criteria in this case.

Kovács & Patakfalvi generalised to log pairs.


What is missing?

We have not given the moduli space a schemestructure.


What is missing?


Equivalently we have not defined a functor.


What is missing?



The problem is that there are families of non-normalvarieties which cannot be smoothed, so that there arecomponents of the moduli space whose generalmember corresponds to a non-normal variety.


What is missing?




These components might intersect the components(the smooth case)whose general points correspondto normal varieties.


What is missing?





We have to include all of the components if we wanta reasonable scheme structure.


What is missing?





We have to include all of the components if we wanta reasonable scheme structure.

Then we get a reasonable deformation theory.Birational geometry and moduli spaces of varieties of general type – p. 18

Surfaces with small invariants

There has been considerable interest in constructingsmooth projective surfaces with pg = 0 and smallfundamental group.




Start with an explicit rational surface S0 withquotient singularities such that KS0

is ample and thefundamental group of the smooth locus is small.






Check that there is a smoothing S of S0 to a smoothprojective surface S such that KS is ample.







Construct new Campedelli surfaces S, smooth

projective surfaces with (Y. Lee, J. Park): K2S = 0

and π1(S) ∈ {0,Z2,Z4}.







Construct new Campedelli surfaces S, smooth

projective surfaces with (Y. Lee, J. Park): K2S = 0

and π1(S) ∈ {0,Z2,Z4}.

Non-trivial deformation theory, Q-Gorensteindeformation (Wahl).


What to parametrise?

A big hint is given by the construction we have justgiven:




• log pairs (X,∆) such that KX +∆ is ample.





We must allow non-normal singularities (considerthe case of nodal curves and what the fibres of asemi-stable reduction look like).






• we work with semi log canonical pairs (X,∆).







X has nodal singularities in codimension one and

the normalisation (Xν, D +∆ν) is log canonical,where D is the double locus.







X has nodal singularities in codimension one and

the normalisation (Xν, D +∆ν) is log canonical,where D is the double locus.

the fibres of the relative canonical model are semilog canonical—part of the statement of adjunction.


Log canonical pairs (X,∆)

(X,∆) is log smooth if X is smooth and ∆ hasglobal normal crossings.




A log resolution π : Y −→ X is a projective

birational map s.t. (Y,Γ = ∆̃ + E) is log smooth

and E =∑

Ei supports an ample divisor over X .






and E =∑


We may write

KY + Γ = π∗(KX +∆) +∑

aiEi.






and E =∑


We may write

KY + Γ = π∗(KX +∆) +∑

aiEi.

ai = a(Ei, X,∆) is the log discrepancy of Ei.






and E =∑


We may write

KY + Γ = π∗(KX +∆) +∑

aiEi.

ai = a(Ei, X,∆) is the log discrepancy of Ei.

a = infi,Y ai is the log discrepancy of (X,∆).(X,∆) is log canonical if a ≥ 0.


Properties of log canonical pairs

If (X,∆) is log canonical then

H0(X,OX(m(KX+∆))) = H0(Y,OY (m(KY+Γ))).

This is the essential property of log canonical pairs.






(X,∆) is klt if ⌊∆⌋ = 0 and a > 0.






(X,∆) is klt if ⌊∆⌋ = 0 and a > 0.

Kawamata log terminal pairs are well-behaved andmost of the standard results about smooth projectivevarieties extend to klt pairs.






(X,∆) is klt if ⌊∆⌋ = 0 and a > 0.


Some of the basic results about log canonical pairsfail and we are ignorant of many of the rest.






(X,∆) is klt if ⌊∆⌋ = 0 and a > 0.


Some of the basic results about log canonical pairsfail and we are ignorant of many of the rest.

We must work with log canonical pairs since thedouble locus always comes with coefficient one.


birational geometry and moduli spaces of varieties of ...though mg is not canonical). birational...

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