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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.
Mircea Mustata (University of Michigan) A singularity invariant in positive characteristicChulalongkorn University December 20, 2011 2
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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.
Mircea Mustata (University of Michigan) A singularity invariant in positive characteristicChulalongkorn University December 20, 2011 2
/ 30
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.
Mircea Mustata (University of Michigan) A singularity invariant in positive characteristicChulalongkorn University December 20, 2011 2
/ 30
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.
Mircea Mustata (University of Michigan) A singularity invariant in positive characteristicChulalongkorn University December 20, 2011 3
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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.
Mircea Mustata (University of Michigan) A singularity invariant in positive characteristicChulalongkorn University December 20, 2011 3
/ 30
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.
Mircea Mustata (University of Michigan) A singularity invariant in positive characteristicChulalongkorn University December 20, 2011 3
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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).
Mircea Mustata (University of Michigan) A singularity invariant in positive characteristicChulalongkorn University December 20, 2011 4
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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
Mircea Mustata (University of Michigan) A singularity invariant in positive characteristicChulalongkorn University December 20, 2011 6
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f = x2 + y2 − z2
mult0(f ) = 2
f = x2 − 3y2z + z3
mult0(f ) = 2
Mircea Mustata (University of Michigan) A singularity invariant in positive characteristicChulalongkorn University December 20, 2011 6
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f = x2 − y2z2
mult0(f ) = 2
f = xy2 + y3 + ...mult0(f ) = 3
Mircea Mustata (University of Michigan) A singularity invariant in positive characteristicChulalongkorn University December 20, 2011 7
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f = x2 − y2z2
mult0(f ) = 2f = xy2 + y3 + ...
mult0(f ) = 3
Mircea Mustata (University of Michigan) A singularity invariant in positive characteristicChulalongkorn University December 20, 2011 7
/ 30
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.
Mircea Mustata (University of Michigan) A singularity invariant in positive characteristicChulalongkorn University December 20, 2011 8
/ 30
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.
Mircea Mustata (University of Michigan) A singularity invariant in positive characteristicChulalongkorn University December 20, 2011 8
/ 30
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.
Mircea Mustata (University of Michigan) A singularity invariant in positive characteristicChulalongkorn University December 20, 2011 8
/ 30
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.
Mircea Mustata (University of Michigan) A singularity invariant in positive characteristicChulalongkorn University December 20, 2011 8
/ 30
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.
Mircea Mustata (University of Michigan) A singularity invariant in positive characteristicChulalongkorn University December 20, 2011 9
/ 30
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.
Mircea Mustata (University of Michigan) A singularity invariant in positive characteristicChulalongkorn University December 20, 2011 9
/ 30
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.
Mircea Mustata (University of Michigan) A singularity invariant in positive characteristicChulalongkorn University December 20, 2011 9
/ 30
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.
Mircea Mustata (University of Michigan) A singularity invariant in positive characteristicChulalongkorn University December 20, 2011 10
/ 30
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.
Mircea Mustata (University of Michigan) A singularity invariant in positive characteristicChulalongkorn University December 20, 2011 10
/ 30
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.
Mircea Mustata (University of Michigan) A singularity invariant in positive characteristicChulalongkorn University December 20, 2011 10
/ 30
A characteristic p invariant, cont’d
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 ) .
Mircea Mustata (University of Michigan) A singularity invariant in positive characteristicChulalongkorn University December 20, 2011 11
/ 30
A characteristic p invariant, cont’d
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 ) .
Mircea Mustata (University of Michigan) A singularity invariant in positive characteristicChulalongkorn University December 20, 2011 11
/ 30
A characteristic p invariant, cont’d
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 ) .
Mircea Mustata (University of Michigan) A singularity invariant in positive characteristicChulalongkorn University December 20, 2011 11
/ 30
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.
Mircea Mustata (University of Michigan) A singularity invariant in positive characteristicChulalongkorn University December 20, 2011 12
/ 30
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.
Mircea Mustata (University of Michigan) A singularity invariant in positive characteristicChulalongkorn University December 20, 2011 12
/ 30
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.
Mircea Mustata (University of Michigan) A singularity invariant in positive characteristicChulalongkorn University December 20, 2011 12
/ 30
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.
Mircea Mustata (University of Michigan) A singularity invariant in positive characteristicChulalongkorn University December 20, 2011 12
/ 30
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.
Mircea Mustata (University of Michigan) A singularity invariant in positive characteristicChulalongkorn University December 20, 2011 13
/ 30
Other properties of F -pure thresholds
Properties:
1) fpt0(f ) ≤ 1.In particular, since fpt0(f ) ≥ 1
mult0(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.
Mircea Mustata (University of Michigan) A singularity invariant in positive characteristicChulalongkorn University December 20, 2011 13
/ 30
Other properties of F -pure thresholds
Properties:
1) fpt0(f ) ≤ 1.In particular, since fpt0(f ) ≥ 1
mult0(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.
Mircea Mustata (University of Michigan) A singularity invariant in positive characteristicChulalongkorn University December 20, 2011 13
/ 30
Other properties of F -pure thresholds
Properties:
1) fpt0(f ) ≤ 1.In particular, since fpt0(f ) ≥ 1
mult0(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.
Mircea Mustata (University of Michigan) A singularity invariant in positive characteristicChulalongkorn University December 20, 2011 13
/ 30
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 ).
Mircea Mustata (University of Michigan) A singularity invariant in positive characteristicChulalongkorn University December 20, 2011 14
/ 30
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 ).
Mircea Mustata (University of Michigan) A singularity invariant in positive characteristicChulalongkorn University December 20, 2011 14
/ 30
Other properties of F -pure thresholds, cont’d
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.
Mircea Mustata (University of Michigan) A singularity invariant in positive characteristicChulalongkorn University December 20, 2011 15
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Other properties of F -pure thresholds, cont’d
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.
Mircea Mustata (University of Michigan) A singularity invariant in positive characteristicChulalongkorn University December 20, 2011 15
/ 30
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.
Mircea Mustata (University of Michigan) A singularity invariant in positive characteristicChulalongkorn University December 20, 2011 16
/ 30
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.
Mircea Mustata (University of Michigan) A singularity invariant in positive characteristicChulalongkorn University December 20, 2011 16
/ 30
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.
Mircea Mustata (University of Michigan) A singularity invariant in positive characteristicChulalongkorn University December 20, 2011 17
/ 30
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.
Mircea Mustata (University of Michigan) A singularity invariant in positive characteristicChulalongkorn University December 20, 2011 17
/ 30
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.
Mircea Mustata (University of Michigan) A singularity invariant in positive characteristicChulalongkorn University December 20, 2011 17
/ 30
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.
Mircea Mustata (University of Michigan) A singularity invariant in positive characteristicChulalongkorn University December 20, 2011 17
/ 30
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).
Mircea Mustata (University of Michigan) A singularity invariant in positive characteristicChulalongkorn University December 20, 2011 18
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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).
Mircea Mustata (University of Michigan) A singularity invariant in positive characteristicChulalongkorn University December 20, 2011 18
/ 30
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).
Mircea Mustata (University of Michigan) A singularity invariant in positive characteristicChulalongkorn University December 20, 2011 18
/ 30
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.
Mircea Mustata (University of Michigan) A singularity invariant in positive characteristicChulalongkorn University December 20, 2011 19
/ 30
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.
Mircea Mustata (University of Michigan) A singularity invariant in positive characteristicChulalongkorn University December 20, 2011 19
/ 30
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.
Mircea Mustata (University of Michigan) A singularity invariant in positive characteristicChulalongkorn University December 20, 2011 19
/ 30
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.
Mircea Mustata (University of Michigan) A singularity invariant in positive characteristicChulalongkorn University December 20, 2011 20
/ 30
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.
Mircea Mustata (University of Michigan) A singularity invariant in positive characteristicChulalongkorn University December 20, 2011 20
/ 30
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.
Mircea Mustata (University of Michigan) A singularity invariant in positive characteristicChulalongkorn University December 20, 2011 20
/ 30
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.
Mircea Mustata (University of Michigan) A singularity invariant in positive characteristicChulalongkorn University December 20, 2011 20
/ 30
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
}.
Mircea Mustata (University of Michigan) A singularity invariant in positive characteristicChulalongkorn University December 20, 2011 21
/ 30
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
}.
Mircea Mustata (University of Michigan) A singularity invariant in positive characteristicChulalongkorn University December 20, 2011 21
/ 30
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
}.
Mircea Mustata (University of Michigan) A singularity invariant in positive characteristicChulalongkorn University December 20, 2011 21
/ 30
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
}.
Mircea Mustata (University of Michigan) A singularity invariant in positive characteristicChulalongkorn University December 20, 2011 21
/ 30
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.
Mircea Mustata (University of Michigan) A singularity invariant in positive characteristicChulalongkorn University December 20, 2011 22
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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.
Mircea Mustata (University of Michigan) A singularity invariant in positive characteristicChulalongkorn University December 20, 2011 22
/ 30
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 .
Mircea Mustata (University of Michigan) A singularity invariant in positive characteristicChulalongkorn University December 20, 2011 23
/ 30
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 .
Mircea Mustata (University of Michigan) A singularity invariant in positive characteristicChulalongkorn University December 20, 2011 23
/ 30
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 .
Mircea Mustata (University of Michigan) A singularity invariant in positive characteristicChulalongkorn University December 20, 2011 23
/ 30
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.
Mircea Mustata (University of Michigan) A singularity invariant in positive characteristicChulalongkorn University December 20, 2011 24
/ 30
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.
Mircea Mustata (University of Michigan) A singularity invariant in positive characteristicChulalongkorn University December 20, 2011 24
/ 30
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.
Mircea Mustata (University of Michigan) A singularity invariant in positive characteristicChulalongkorn University December 20, 2011 24
/ 30
F -pure threshold vs. log canonical threshold, cont’d
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.
Mircea Mustata (University of Michigan) A singularity invariant in positive characteristicChulalongkorn University December 20, 2011 25
/ 30
F -pure threshold vs. log canonical threshold, cont’d
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.
Mircea Mustata (University of Michigan) A singularity invariant in positive characteristicChulalongkorn University December 20, 2011 25
/ 30
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 ).
Mircea Mustata (University of Michigan) A singularity invariant in positive characteristicChulalongkorn University December 20, 2011 26
/ 30
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 ).
Mircea Mustata (University of Michigan) A singularity invariant in positive characteristicChulalongkorn University December 20, 2011 26
/ 30
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 ).
Mircea Mustata (University of Michigan) A singularity invariant in positive characteristicChulalongkorn University December 20, 2011 26
/ 30
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 ).
Mircea Mustata (University of Michigan) A singularity invariant in positive characteristicChulalongkorn University December 20, 2011 26
/ 30
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 .
Mircea Mustata (University of Michigan) A singularity invariant in positive characteristicChulalongkorn University December 20, 2011 27
/ 30
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 .
Mircea Mustata (University of Michigan) A singularity invariant in positive characteristicChulalongkorn University December 20, 2011 27
/ 30
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 .
Mircea Mustata (University of Michigan) A singularity invariant in positive characteristicChulalongkorn University December 20, 2011 27
/ 30
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
Mircea Mustata (University of Michigan) A singularity invariant in positive characteristicChulalongkorn University December 20, 2011 28
/ 30
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
Mircea Mustata (University of Michigan) A singularity invariant in positive characteristicChulalongkorn University December 20, 2011 28
/ 30
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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/ 30
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.
Mircea Mustata (University of Michigan) A singularity invariant in positive characteristicChulalongkorn University December 20, 2011 29
/ 30
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.
Mircea Mustata (University of Michigan) A singularity invariant in positive characteristicChulalongkorn University December 20, 2011 29
/ 30
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.
Mircea Mustata (University of Michigan) A singularity invariant in positive characteristicChulalongkorn University December 20, 2011 29
/ 30
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.
Mircea Mustata (University of Michigan) A singularity invariant in positive characteristicChulalongkorn University December 20, 2011 30
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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.
Mircea Mustata (University of Michigan) A singularity invariant in positive characteristicChulalongkorn University December 20, 2011 30
/ 30
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.
Mircea Mustata (University of Michigan) A singularity invariant in positive characteristicChulalongkorn University December 20, 2011 30
/ 30
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.
Mircea Mustata (University of Michigan) A singularity invariant in positive characteristicChulalongkorn University December 20, 2011 30
/ 30