Mircea Mustat˘ a - University of Cambridgecb496/conf-chula/talk-mustata-1.pdf · A singularity invariant in positive characteristic Mircea Mustat˘ a University of Michigan Chulalongkorn

A singularity invariant in positive characteristic

Mircea Mustata

University of Michigan

Chulalongkorn UniversityDecember 20, 2011

Mircea Mustata (University of Michigan) A singularity invariant in positive characteristicChulalongkorn University December 20, 2011 1

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Singular points

Suppose U ⊂ Rn is an open subset, f is a C∞ function on U.This defines the hypersurface H = {x ∈ U | f (x) = 0}.

A point q ∈ H is a nonsingular or smooth point of f if some ∂f∂xi

(q) 6= 0.In this case, H is a submanifold of U. Otherwise, the point is singular.

In algebraic geometry: work with Cn and f ∈ C[x1, . . . , xn] a nonzeropolynomial.

f =∑α∈Zn

≥0

cαxα11 . . . xαn

n finite sum, with cα ∈ C.

For simplicity: assume q = 0. Note: 0 ∈ H ⇔ c0 = 0.

0 ∈ H is a smooth point iff f contains some monomial of degree one.


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Singular points





f =∑α∈Zn

≥0

cαxα11 . . . xαn





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Singular points





f =∑α∈Zn

≥0

cαxα11 . . . xαn





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Idea: all smooth points are alike (close to such a point, H looks likeCn−1), but each singular point is singular in its own way.

Goal: describe some invariants that measure the singularities.

Most basic invariant: the multiplicity. One defines mult0(f ) to be thesmallest degree of a monomial that appears in f .

To keep in mind:

1) mult0(f ) = 1 iff 0 is a smooth point.

2) “Worse” singularities have larger multiplicity.

3) Very rough invariant: often polynomials with the same multiplicityhave very different singularities.


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To keep in mind:





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To keep in mind:





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Examples of singularities (pictures due to R. Lazarsfeld,H. Hauser, and O. Labs)

f = y − x2 , mult0(f ) = 1 (smooth point).


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Examples, cont’d

f = 2x3 + 3x2 − 3y2

mult0(f ) = 2

f = (x2 + y2)2 + 3x2y − y3

mult0(f ) = 3


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f = x2 + y2 − z2

mult0(f ) = 2

f = x2 − 3y2z + z3

mult0(f ) = 2


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f = x2 + y2 − z2

mult0(f ) = 2

f = x2 − 3y2z + z3

mult0(f ) = 2


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f = x2 − y2z2

mult0(f ) = 2

f = xy2 + y3 + ...mult0(f ) = 3


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f = x2 − y2z2

mult0(f ) = 2f = xy2 + y3 + ...

mult0(f ) = 3


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New setting: positive characteristic

Let k be a field of positive characteristic p. Main examples:

1) Fp = Z/pZ

2) Finite extension Fq of Fp, with q = pe

3) The algebraic closure Fp of Fp

Let R = k[x1, . . . , xn] and f ∈ R nonzero, with f (0) = 0.Definition of singular points and of multiplicity is the same as over C.

Unpleasant feature in characteristic p:∂(xpi )∂xi

= 0.

Miracle: (a + b)p = ap + bp since(pi

)= 0 in k for 0 < i < p.

Upshot: F : R → R given by F (h) = hp is a ring homomorphism (theFrobenius homomorphism). This is responsible for the interestingphenomena in positive characteristic.


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1) Fp = Z/pZ





= 0.


)= 0 in k for 0 < i < p.



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1) Fp = Z/pZ





= 0.


)= 0 in k for 0 < i < p.



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1) Fp = Z/pZ





= 0.


)= 0 in k for 0 < i < p.



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A characteristic p invariant

We reinterpret the definition of multiplicity. Let

m = m0 = (x1, . . . , xn).

The power md is generated by the monomials of degree d . It consists ofall polynomials of multiplicity ≥ d (at 0).

Easy to see: mult0(f i ) = i ·mult0(f ).

In particular, f i ∈ md if and only if i ·mult0(f ) ≥ d .

It is much more interesting to ask which powers of f lie in the Frobeniuspowers of m.


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m = m0 = (x1, . . . , xn).






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m = m0 = (x1, . . . , xn).






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A characteristic p invariant, cont’d

The Frobenius eth-power of m is

m[pe ] = (hpe | h ∈ m) = (xp

e

1 , . . . , xpe

n ).

For every e ≥ 1, we put

ν(e) = νf (pe) := smallest r ≥ 0 such that f r ∈ m[pe ].

Note:

1) ν(e) ≤ pe .

2) If f r ∈ (xpe

1 , . . . , xpe

n ), then f pr ∈ (xpe+1

1 , . . . , xpe+1

r ). Therefore

ν(e + 1)

pe+1≤ ν(e)

pe.


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m[pe ] = (hpe | h ∈ m) = (xp

e

1 , . . . , xpe

n ).



Note:

1) ν(e) ≤ pe .

2) If f r ∈ (xpe

1 , . . . , xpe


1 , . . . , xpe+1

r ). Therefore

ν(e + 1)

pe+1≤ ν(e)

pe.


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m[pe ] = (hpe | h ∈ m) = (xp

e

1 , . . . , xpe

n ).



Note:

1) ν(e) ≤ pe .

2) If f r ∈ (xpe

1 , . . . , xpe


1 , . . . , xpe+1

r ). Therefore

ν(e + 1)

pe+1≤ ν(e)

pe.


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The F -pure threshold of f (at 0) is

fpt0(f ) := infe≥1

ν(e)

pe= lim

e→∞

ν(e)

pe.

Introduced by Takagi and Watanabe in 2004 (with a different definition, inthe context of tight closure theory).

If we play the same game with usual powers instead of Frobenius powers:

ν ′(`) = smallest integer r such that f r ∈ m`

= d`/mult0(f )e,

hence lim`→∞ν′(`)` = lim`→∞

d`/mult0(f )e` = 1

mult0(f ) .


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ν(e)

pe= lim

e→∞

ν(e)

pe.




= d`/mult0(f )e,


d`/mult0(f )e` = 1

mult0(f ) .


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ν(e)

pe= lim

e→∞

ν(e)

pe.




= d`/mult0(f )e,


d`/mult0(f )e` = 1

mult0(f ) .


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F -pure threshold vs. multiplicity

Note that for every e ≥ 1, we have the following inclusions of ideals

mn(pe−1)+1 ⊆ m[pe ] ⊆ mpe .

This gives the inequalities

ν ′(pe) ≤ ν(e) ≤ ν ′(n(pe − 1) + 1).

Dividing by pe , and letting e →∞, we obtain

1

mult0(f )≤ fpt0(f ) ≤ n

mult0(f ).

Conclusion: fpt0(f ) behaves like 1mult0(f ) ; we will see, however, that it is a

more subtle invariant.


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mn(pe−1)+1 ⊆ m[pe ] ⊆ mpe .


ν ′(pe) ≤ ν(e) ≤ ν ′(n(pe − 1) + 1).


1


mult0(f ).




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mn(pe−1)+1 ⊆ m[pe ] ⊆ mpe .


ν ′(pe) ≤ ν(e) ≤ ν ′(n(pe − 1) + 1).


1


mult0(f ).




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mn(pe−1)+1 ⊆ m[pe ] ⊆ mpe .


ν ′(pe) ≤ ν(e) ≤ ν ′(n(pe − 1) + 1).


1


mult0(f ).




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Other properties of F -pure thresholds

Properties:

1) fpt0(f ) ≤ 1.

In particular, since fpt0(f ) ≥ 1mult0(f ) , we conclude that if 0 is a

nonsingular point of f , then fpt0(f ) = 1.

2) fpt0(f m) = fpt0(f )m .

Proof. (f m)r ∈ m[pe ] iff mr ≥ νf (e).Therefore νf m(e) = dνf (e)/me and

lime→∞

νf m(e)

pe= lim

e→∞

dνf (e)/mepe

=fpt0(f )

m.


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Properties:

1) fpt0(f ) ≤ 1.In particular, since fpt0(f ) ≥ 1

mult0(f ) , we conclude that if 0 is a


2) fpt0(f m) = fpt0(f )m .


lime→∞

νf m(e)

pe= lim

e→∞

dνf (e)/mepe

=fpt0(f )

m.


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Properties:




2) fpt0(f m) = fpt0(f )m .

Proof. (f m)r ∈ m[pe ] iff mr ≥ νf (e).

Therefore νf m(e) = dνf (e)/me and

lime→∞

νf m(e)

pe= lim

e→∞

dνf (e)/mepe

=fpt0(f )

m.


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Properties:




2) fpt0(f m) = fpt0(f )m .


lime→∞

νf m(e)

pe= lim

e→∞

dνf (e)/mepe

=fpt0(f )

m.


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Other properties of F -pure thresholds, cont’d

3) Behavior under restriction:If g = f (x1, . . . , xn−1, 0) ∈ R ′ = k[x1, . . . , xn−1], then

fpt0(g) ≤ fpt0(f )

(that is, g has worse singularities than f , from the point of view ofthe F -pure threshold)

Proof. We put m′ = (x1, . . . , xn−1) ⊆ S .

Clear: if f r ∈ m[pe ], then g r ∈ m′[pe ], hence

νg (e) ≤ νf (e).

Dividing by pe and letting e →∞ gives fpt0(g) ≤ fpt0(f ).


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3) Behavior under restriction:If g = f (x1, . . . , xn−1, 0) ∈ R ′ = k[x1, . . . , xn−1], then

fpt0(g) ≤ fpt0(f )

(that is, g has worse singularities than f , from the point of view ofthe F -pure threshold)Proof. We put m′ = (x1, . . . , xn−1) ⊆ S .

Clear: if f r ∈ m[pe ], then g r ∈ m′[pe ], hence

νg (e) ≤ νf (e).

Dividing by pe and letting e →∞ gives fpt0(g) ≤ fpt0(f ).


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4) Behavior with respect to sum: if f1, f2 as above, then

fpt0(f1 + f2) ≤ fpt0(f1) + fpt0(f2).

Proof. If f r1 ∈ m[pe ] and f s2 ∈ m[pe ], then

(f1 + f2)r+s ∈ m[pe ].

Therefore νf1+f2(e) ≤ νf1(e) + νf2(e).

Dividing by pe , and letting e →∞ gives what we want.


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4) Behavior with respect to sum: if f1, f2 as above, then

fpt0(f1 + f2) ≤ fpt0(f1) + fpt0(f2).

Proof. If f r1 ∈ m[pe ] and f s2 ∈ m[pe ], then

(f1 + f2)r+s ∈ m[pe ].

Therefore νf1+f2(e) ≤ νf1(e) + νf2(e).

Dividing by pe , and letting e →∞ gives what we want.


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Rationality

Theorem (Blickle,M-, Smith; 2008). If f is as above, then

fpt0(f ) ∈ Q.

This comes from a “finiteness” statement. The proof is more involved, sowe do not discuss it here.


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Rationality

Theorem (Blickle,M-, Smith; 2008). If f is as above, then

fpt0(f ) ∈ Q.

This comes from a “finiteness” statement. The proof is more involved, sowe do not discuss it here.


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Examples of F -pure thresholds: monomials

1) If f = xa11 · · · xann , then

fpt0(f ) = mini

{1

ai

}.

Proof.f r = x ra1

1 · · · x rann ∈ m[pe ] iff rai ≥ pe for some i .It follows that if d = maxi ai , then ν(e) = dpe/de, hence

lime→∞

ν(e)

pe= lim

e→∞

dpe/depe

=1

d.

Note: fpt0(x1 · · · xi ) = 1, hence the F -pure threshold can be 1 alsowhen 0 is a singular point.


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fpt0(f ) = mini

{1

ai

}.

Proof.f r = x ra1

1 · · · x rann ∈ m[pe ] iff rai ≥ pe for some i .

It follows that if d = maxi ai , then ν(e) = dpe/de, hence

lime→∞

ν(e)

pe= lim

e→∞

dpe/depe

=1

d.



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fpt0(f ) = mini

{1

ai

}.

Proof.f r = x ra1


lime→∞

ν(e)

pe= lim

e→∞

dpe/depe

=1

d.



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fpt0(f ) = mini

{1

ai

}.

Proof.f r = x ra1


lime→∞

ν(e)

pe= lim

e→∞

dpe/depe

=1

d.



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Examples of F -pure thresholds: the cusp

2) fp = x2 + y3 ∈ Fp[x , y ], hence mult0(fp) = 2.

Let us compute ν(1). If r ≤ p − 1, then

f rp ∈ (xp, yp) iff the following holds :

for all i , j ≥ 0 with i + j = r , either 2i ≥ p or 3j ≥ p.It follows that

ν(1) = b(p − 1)/2c+ b(p − 1)/3c+ 1 =

1, for p = 2 or 3;

5p+16 , for p ≡ 1 (mod 3);

5p−16 , for p ≡ 2 (mod 3).


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for all i , j ≥ 0 with i + j = r , either 2i ≥ p or 3j ≥ p.

It follows that

ν(1) = b(p − 1)/2c+ b(p − 1)/3c+ 1 =

1, for p = 2 or 3;

5p+16 , for p ≡ 1 (mod 3);

5p−16 , for p ≡ 2 (mod 3).


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for all i , j ≥ 0 with i + j = r , either 2i ≥ p or 3j ≥ p.It follows that

ν(1) = b(p − 1)/2c+ b(p − 1)/3c+ 1 =

1, for p = 2 or 3;

5p+16 , for p ≡ 1 (mod 3);

5p−16 , for p ≡ 2 (mod 3).


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Examples of F -pure thresholds: the cusp, cont’d

One can compute all ν(e) by induction on e ≥ 1. One obtains thefollowing

fpt0(fp) =

12 , for p = 2;

23 , forp = 3;

56 , for p ≡ 1 (mod 3);

56 −

16p , for p ≡ 2 (mod 3), p > 2.

The singularity of fp is the same in any characteristic; we see, however, anarithmetic contribution to fpt0(fp) from the characteristic of k .

To keep in mind: we have c = 56 such that

• c ≥ fpt0(fp) for all p.• limp→∞ fpt0(fp) = c• fpt0(fp) = c for infinitely many p.


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fpt0(fp) =

12 , for p = 2;

23 , forp = 3;

56 , for p ≡ 1 (mod 3);

56 −

16p , for p ≡ 2 (mod 3), p > 2.





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fpt0(fp) =

12 , for p = 2;

23 , forp = 3;

56 , for p ≡ 1 (mod 3);

56 −

16p , for p ≡ 2 (mod 3), p > 2.





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A general question

Suppose now that f ∈ Z[x1, . . . , xn] is a nonzero polynomial such thatf (0) = 0.

For every prime p, we obtain by reduction mod p a polynomialfp ∈ Fp[x1, . . . , xn].

A vague question. How do the invariants fpt0(fp) vary with p? Forexample, in the case of the cusp, can we read off 5

6 from the characteristiczero polynomial?

In general, we may consider f also as a polynomial in C[x1, . . . , xn], andstudy its invariants defined in this characteristic zero setting.


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A general question







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A general question







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A general question







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The log canonical threshold

Let f ∈ C[x1, . . . , xn] with f (0) = 0.

The integrability exponent, also called log canonical threshold in birationalgeometry is

lct0(f ) = sup

{s > 0 | 1

|f (x)|2sis integrable around 0

}.

It was introduced and first studied by Varchenko in the early 80’s. It hashad an important role in the birational study of higher-dimensionalalgebraic varieties, starting with the work of Shokurov in 1992.

Example. If f = xa11 · · · xann , then

lct0(f ) = mini

{1

ai

}.


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Let f ∈ C[x1, . . . , xn] with f (0) = 0.


lct0(f ) = sup

{s > 0 | 1


}.



lct0(f ) = mini

{1

ai

}.


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Let f ∈ C[x1, . . . , xn] with f (0) = 0.


lct0(f ) = sup

{s > 0 | 1


}.



lct0(f ) = mini

{1

ai

}.


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Let f ∈ C[x1, . . . , xn] with f (0) = 0.


lct0(f ) = sup

{s > 0 | 1


}.



lct0(f ) = mini

{1

ai

}.


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The log canonical threshold, cont’d

Indeed, using Fubini’s Theorem, we can assume n = 1. In this case, thechange of variable

x = r · cos(θ), y = r · sin(θ), 0 ≤ θ ≤ 2π

gives ∫x2+y2<ε

dx dy

(x2 + y2)ai s= 2π

∫ √ε0

r1−2ai sdr <∞

if and only if 1− 2ai s > −1, that is, s < 1ai

.

The log canonical threshold can be computed also algebraically using aversion of resolution of singularities, which exists by a fundamentaltheorem of Hironaka. Vague idea: this consists of a finite set of algebraicchanges of variables, that reduce us to the case when

f = xa11 · · · x

ann ,

involving also the Jacobians of these changes of variables.Upshot: lct0(f ) ∈ Q.


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The log canonical threshold, cont’d

Indeed, using Fubini’s Theorem, we can assume n = 1. In this case, thechange of variable

x = r · cos(θ), y = r · sin(θ), 0 ≤ θ ≤ 2π

gives ∫x2+y2<ε

dx dy

(x2 + y2)ai s= 2π

∫ √ε0

r1−2ai sdr <∞

if and only if 1− 2ai s > −1, that is, s < 1ai

.

The log canonical threshold can be computed also algebraically using aversion of resolution of singularities, which exists by a fundamentaltheorem of Hironaka. Vague idea: this consists of a finite set of algebraicchanges of variables, that reduce us to the case when

f = xa11 · · · x

ann ,

involving also the Jacobians of these changes of variables.Upshot: lct0(f ) ∈ Q.


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F -pure threshold vs. log canonical threshold

Part of the motivation of Takagi and Watanabe for introducing the F -purethreshold was that it satisfied properties similar to those satisfied by thelog canonical threshold.

In fact, all properties that we discussed had been known for log canonicalthresholds (though the proofs are more involved).

Example If f = xa11 + . . .+ xann ¡ then

lct0(f ) = min

{n∑

i=1

1

ai, 1

}.

In particular, lct0(x2 + y3) = 12 + 1

3 = 56 .


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lct0(f ) = min

{n∑

i=1

1

ai, 1

}.


3 = 56 .


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lct0(f ) = min

{n∑

i=1

1

ai, 1

}.


3 = 56 .


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F -pure threshold vs. log canonical threshold, cont’d

Theorem (Hara-Yoshida, 2003). If f ∈ Z[x1, . . . , xn] is nonzero, withf (0) = 0, then the following hold:

i) For all primes p � 0, we have

lct0(f ) ≥ fpt0(fp).

ii) We have limp→∞ fpt0(fp) = lct0(f ).

The proof of part i) is not hard, once the right machinery is set into place.The proof of ii) is deeper, making use of results about the Frobeniusaction on the de Rham complex.


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The main open problem in this area is the following:Conjecture. If f ∈ Z[x1, . . . , xn] is nonzero, with f (0) = 0, then there isan infinite set S of primes such that

fpt0(fp) = lct0(f ) for all p ∈ S .

Remark. One expects that the set of primes S in the conjecture haspositive density. For example, in the case of the cusp, it has density 1

2 . Infact, at least a subset of this S should have an arithmetic interpretation.


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The main open problem in this area is the following:Conjecture. If f ∈ Z[x1, . . . , xn] is nonzero, with f (0) = 0, then there isan infinite set S of primes such that

fpt0(fp) = lct0(f ) for all p ∈ S .

Remark. One expects that the set of primes S in the conjecture haspositive density. For example, in the case of the cusp, it has density 1

2 . Infact, at least a subset of this S should have an arithmetic interpretation.


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An example: cones over Calabi-Yau hypersurfaces

Let us consider the following example.

Let f ∈ Z[x1, . . . , xn] be homogeneous of degree n ≥ 3, so mult0(f ) = n.Since f is homogeneous, C∗ acts on {u ∈ Cn r {0} | f (u) = 0} byrescaling the coordinates, and the quotient is the projective hypersurface Zdefined by f .

We assume that the only singular point of f (in Cn) is 0, which meansthat all points of Z are smooth. When n = 3, Z has (complex) dimensionone. It is an elliptic curve.

Easy to see: in general we have lct0(f ) = 1(consider, for example, the case f = xn1 + . . .+ xnn ).


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An example: cones over Calabi-Yau hypersurfaces, cont’d

What is the F -pure threshold of fp?

Note that f p−1p is a homogeneous polynomial of degree n(p − 1). We can

write

f p−1p = λp(x1 · · · xn)p−1 + gp, with λp ∈ Fp and gp ∈ (xp1 , . . . , x

pn ).

For e ≥ 1, we have

f pe−1

p =e−1∏i=0

f pi (p−1) =

e−1∏i=0

(λp

i

p (x1 · · · xn)pi (p−1) + gpi

p

)

≡ λpe−1p−1p (x1 · · · xn)p

e−1 (mod (xpe

1 , . . . , xpe

n )).

If λp 6= 0, then ν(e) ≥ pe for all e, hence fpt0(f ) ≥ 1.If λp = 0, then ν(e) ≤ p − 1, hence fpt0(f ) ≤ 1− 1

p .


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write


pn ).


f pe−1

p =e−1∏i=0

f pi (p−1) =

e−1∏i=0

(λp

i

p (x1 · · · xn)pi (p−1) + gpi

p

)

≡ λpe−1p−1p (x1 · · · xn)p

e−1 (mod (xpe

1 , . . . , xpe

n )).


p .


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write


pn ).


f pe−1

p =e−1∏i=0

f pi (p−1) =

e−1∏i=0

(λp

i

p (x1 · · · xn)pi (p−1) + gpi

p

)

≡ λpe−1p−1p (x1 · · · xn)p

e−1 (mod (xpe

1 , . . . , xpe

n )).


p .


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An example: cones over elliptic curves

When n = 3 (cone over an elliptic curve), the two cases have been studiedfor a long time for independent reasons. Suppose fp has only one singularpoint, at 0.• When λp = 0, the elliptic curve mod p is supersingular.• When λp 6= 0, the elliptic curve mod p is ordinary.

Two concrete examples. The ones in blue are the ordinary ones, and thosefor which fp has a singularity outside 0 appear in parentheses

1) f = y2z + yz2 − x3 + 7z3

2, (3), 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53,59, 61, 67, 71, 73, 79, 83, 89, 97

2) f = y z + xyz + yz2 − x3 + x2z + xz2 + 14z3

2, 3, 5, 7, 11, 13, (17), 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71,73, 79, 83, 89, 97


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1) f = y2z + yz2 − x3 + 7z3

2, (3), 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53,59, 61, 67, 71, 73, 79, 83, 89, 97

2) f = y z + xyz + yz2 − x3 + x2z + xz2 + 14z3

2, 3, 5, 7, 11, 13, (17), 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71,73, 79, 83, 89, 97


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1) f = y2z + yz2 − x3 + 7z3

2, (3), 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53,59, 61, 67, 71, 73, 79, 83, 89, 97

2) f = y z + xyz + yz2 − x3 + x2z + xz2 + 14z3

2, 3, 5, 7, 11, 13, (17), 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71,73, 79, 83, 89, 97


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An example: cones over elliptic curves, cont’d

In general, the elliptic curves are divided into two groups, as follows:

1) Elliptic curves as the first one above, that have more symmetries thanexpected (have complex multiplication). Then half of the primes areordinary.

2) Elliptic curves with as few symmetries as possible (without complexmultiplication; these are the general ones). Then it is known that theset of ordinary primes has density one (Serre), but there are infinitelymany supersingular primes (Elkies).

Conclusion: this solves the conjecture for cones over elliptic curves.


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Cones over Calabi-Yau hypersurfaces: conclusion

As we have seen, the conjecture admits an elementary interpretation in thecase of cones over Calabi-Yau hypersurfaces (f ∈ k[x1, . . . , xn],homogeneous of degree n). However, it seems quite hard to attack theconjecture using this interpretation.

In the case of cones over elliptic curves (n = 3), the conjecture followsfrom the extensive knowledge about elliptic curves.

The case n = 4 can also be solved (though less explicitly), using somedeeper methods.

The conjecture is wide open starting with n ≥ 5, and also for f ∈ k[x , y , z ]homogeneous of degree ≥ 4.


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Mircea Mustat˘ a - University of Cambridgecb496/conf-chula/talk-mustata-1.pdf · A singularity invariant in positive characteristic Mircea Mustat˘ a University of Michigan Chulalongkorn