Real lines on random cubic surfaces Chiara Meroni ICERM August 28, - - PowerPoint PPT Presentation
Real lines on random cubic surfaces Chiara Meroni ICERM August 28, 2020 August 28, 2020 1 / 26 Chiara Meroni Real lines on random cubic surfaces joint work with Rida Ait El Manssour and Mara Belotti based on [BLLP]: Saugata Basu, Antonio
Chiara Meroni ICERM August 28, 2020
Chiara Meroni Real lines on random cubic surfaces August 28, 2020 1 / 26
joint work with Rida Ait El Manssour and Mara Belotti based on [BLLP]: Saugata Basu, Antonio Lerario, Erik Lundberg, and Chris Peterson. Random fields and the enumerative geometry of lines on real and complex
Chiara Meroni Real lines on random cubic surfaces August 28, 2020 2 / 26
Introduction
Let K = R or C and consider f ∈ K[x0, x1, x2, x3](3) Definition The cubic surface over the field K associated to f is Z(f ) = {[x0, x1, x2, x3] ∈ KP3 | f (x0, x1, x2, x3) = 0}. Z(f ) ⊂ KP3 is smooth for the generic choice of f . Question: How many lines are there on a generic cubic surface?
Chiara Meroni Real lines on random cubic surfaces August 28, 2020 3 / 26
Introduction
Classical algebraic geometry: Theorem (Cayley, Salmon - 1849) Every smooth cubic surface over an algebraically closed field contains exactly 27 lines.
Figure: Cubic surface with 27 lines
Chiara Meroni Real lines on random cubic surfaces August 28, 2020 4 / 26
Introduction
No generic answer! Schläfli (1863) The number of lines on a real smooth cubic surface is 27, 15, 7 or 3. The idea is to substitute the word “generic” with the word “random” and ask Question updated: What is the expected number of lines on a random cubic surface?
Chiara Meroni Real lines on random cubic surfaces August 28, 2020 5 / 26
Introduction
Put a probability distribution on R[x0, x1, x2, x3](3). Requirements: centered gaussian O(4)-invariant classify such distributions
classify O(4)-invariant scalar products
Chiara Meroni Real lines on random cubic surfaces August 28, 2020 6 / 26
Harmonic decomposition
More in general let Wn,d = R[x0, . . . xn](d); O(n + 1) acts on Wn,d Aim Find the decomposition of Wn,d into its irreducible subrepresentations. Definition Define Hn
d := {H ∈ Wn,d : ∆H = 0}
to be the space of real homogeneous harmonic polynomials of degree d in n + 1 variables.
Chiara Meroni Real lines on random cubic surfaces August 28, 2020 7 / 26
Harmonic decomposition
Decomposition Wn,d =
xd−jHn
j
Each Hn
j is O(n + 1)-invariant and irreducible
The decomposition is orthogonal w.r.t. every invariant scalar product Every invariant scalar product on Hn
j is a multiple of the L2 scalar
product Given f , g ∈ Wn,d we can write f =
j xd−jfj and g = j xd−jgj,
with fj, gj ∈ Hn
j , and we have that
(f , g) =
µj(fj, gj)2 for some µd, µd−2, . . . > 0.
Chiara Meroni Real lines on random cubic surfaces August 28, 2020 8 / 26
Harmonic decomposition
Fix a harmonic basis {Hj,i} ⊂ Hn
j , orthonormal with respect to (·, ·)2; then
{
1 √µj Hj,i} is an orthonormal basis of Wn,d with respect to (·, ·).
Random polynomial: P(x) =
λj
ξj,ixd−jHj,i(x) ξj,i ∼ N(0, 1) where Jj = dim(Hn
j ).
Chiara Meroni Real lines on random cubic surfaces August 28, 2020 9 / 26
Harmonic decomposition
W3,3 = H3
3 ⊕ x2H3 1
Assume that λ1 + λ3 = 1, then Pλ(x) = λ
j∈J3
ξ3,j · H3,j(x)
+ (1 − λ) x2
j∈J1
ξ1,j · H1,j(x)
where ξi,j ∼ N(0, 1) for all i, j and independent. The distributions we are interested in can be parametrized by the single scalar λ ∈ [0, 1].
Chiara Meroni Real lines on random cubic surfaces August 28, 2020 10 / 26
Kostlan distribution - [BLLP]
In particular for λ = 1
3 we get the Kostlan polynomial:
P(x) = P 1
3 (x) =
ξα ·
α0! · · · α3! xα0
0 · · · xα3 3
Theorem (Basu, Lerario, Lundberg, Peterson) The expected number E of real lines on a random Kostlan cubic surface in RP3 is E = 6 √ 2 − 3.
Chiara Meroni Real lines on random cubic surfaces August 28, 2020 11 / 26
Kostlan distribution - [BLLP]
(1) The Grassmannian. Let Gr(2, 4) denote the Grassmannian of 2-planes in R4, and let sym3(τ ∗
2,4) be the 3rd symmetric power of the cotangent of the
tautological bundle on Gr(2, 4). Every f ∈ R[x0, x1, x2, x3](3) defines a section σf of the bundle sym3(τ ∗
2,4):
σf (W ) = f |W . sym3(τ ∗
2,4)
Gr(2, 4)
π σf
The problem of finding the expected number of lines in the surface Z(P) ⊆ RP3 becomes computing E = E#{W ∈ Gr(2, 4) | σP(W ) = 0}.
Chiara Meroni Real lines on random cubic surfaces August 28, 2020 12 / 26
Kostlan distribution - [BLLP]
(2) Trivialization of the bundle and Kac-Rice formula. Theorem (Kac-Rice formula) Let U ⊂ RN be an open set and X : U → RN be a random map such that: X is gaussian; X is almost surely of class C1; for every t ∈ U the random variable X(t) has a nondegenerate distribution; the probability that X has degenerate zeroes in U is zero; Then, denoting by pX(t) the density function of X(t), for every Borel subset B ⊂ U we have: E#({X = 0} ∩ B) =
E{|det(JX(t))| | X(t) = 0}pX(t)(0)dt where JX(t) denotes the Jacobian matrix of X(t).
Chiara Meroni Real lines on random cubic surfaces August 28, 2020 13 / 26
Kostlan distribution - [BLLP]
Kac-Rice formula + trivialization of the (oriented) Grassmannian and its vector bundle
E#{˜ σP = 0} =
E{|det(J(W ))| | ˜ σP(W ) = 0}p(0, W ) · wGr+(2,4)(W )
Chiara Meroni Real lines on random cubic surfaces August 28, 2020 14 / 26
Kostlan distribution - [BLLP]
FACTS: E{|det(J(W ))| | ˜ σP(W ) = 0}p(0, W ) is a constant that does not depend on W , because P is O(4) invariant J(W ) and ˜ σP(W ) are independent E = E#{W ∈ Gr(2, 4) | σPλ(W ) = 0} = E{|det(J(W0))|} · p(0, W0) · vol(Gr(2, 4)) where W0 = {x2 = 0, x3 = 0}.
Chiara Meroni Real lines on random cubic surfaces August 28, 2020 15 / 26
Kostlan distribution - [BLLP]
(3) E{|det(J(W0))|}. Up to a constant the matrix J(W0) is ˆ J =
a d √ 2b a √ 2e d c √ 2b f √ 2e c f
where a, b, c, d, e, f ∼ N(0, 1). x = bf − ce y = af − cd z = ae − bd
J )| = |2xz − y2|
Chiara Meroni Real lines on random cubic surfaces August 28, 2020 16 / 26
Kostlan distribution - [BLLP]
Finally E{|det(ˆ J)|} = 1 4π
|(x, y, z)|dxdydz and so E = 6 √ 2 − 3
Chiara Meroni Real lines on random cubic surfaces August 28, 2020 17 / 26
Invariant distributions
Theorem (Ait El Manssour, Belotti, M.) The expected number of real lines on the zero set of the random cubic polynomial Pλ equals: Eλ = 9(8λ2 + (1 − λ)2) 2λ2 + (1 − λ)2
8λ2 + (1 − λ)2 − 1 3 + 2 3
20λ2 + (1 − λ)2
Chiara Meroni Real lines on random cubic surfaces August 28, 2020 18 / 26
Invariant distributions
(1) The Grassmannian: as the Kostlan case (2) Trivialization of the bundle and Kac-Rice formula: as the Kostlan case so Eλ = E{|det(J(W0))|}(λ) · p(0, W0)(λ) · vol(Gr(2, 4)) (3) E{|det(J(W0))|}: the key is to find the correct harmonic basis! Pλ(x) = λ
j∈J3
ξ3,j · H3,j(x)
+ (1 − λ)
j∈J1
ξ1,j · x2H1,j(x)
Chiara Meroni Real lines on random cubic surfaces August 28, 2020 19 / 26
Invariant distributions
J(W0) =
a − b d − e c a − b f d − e a + b c d + e f a + b d + e
where a ∼ d, b ∼ e, c ∼ f and their variance is a function of λ. E{|det(J(W0))|} = = 1 4π
6 + (1 − λ)2 48
4 y2 +
6 + (1 − λ)2 48
· e−|(x,y,z)| |(x, y, z)| dxdydz
Chiara Meroni Real lines on random cubic surfaces August 28, 2020 20 / 26
Invariant distributions
1/3 2/3 1 1 2 3 4 5 6 7 8 9 10 11 12 13 Maximum: E1 = 24
5 − 3 ≃ 12, 179
The value λ = 1 corresponds to purely harmonic polynomials of degree 3.
Chiara Meroni Real lines on random cubic surfaces August 28, 2020 21 / 26
Generalizations p-adics
f ∈ Qp[x0, x1, x2, x3](3), Z(f ) ⊂ QpP3, GL4(Zp) acts on the space Qp[x0, x1, x2, x3](3); P(x) =
ξα xα0
0 · · · xα3 3
(1) The Grassmannian (2) Trivialization of the bundle and Kac-Rice formula E = lim
m→+∞
µ(GrQp (2,4)) λ(Bm) µ(Am)
·
E{| det(Jψ(t))|p | ψ(t)=0}·pψ(t)(0) dt λ(Bm)
=
λ(GL4(Zp)) λ(GL2(Zp)) λ(GL2(Zp)) · E
=
λ(GL4(Zp)) λ(GL2(Zp)) · λ(GL2(Zp)) ·E{| det(Jψ(0))|p} Chiara Meroni Real lines on random cubic surfaces August 28, 2020 22 / 26
Generalizations p-adics
(3) E {| det(Jψ(0))|p} M =
a d b a e d c b f e c f
and a, b, c, d, e, f are random variables i.i.d uniformly distributed in Zp. E{| det(M)|p} = lim
n→∞
1 p6n
|det(M)|p Theorem (Ait El Manssour) The expected number of lines on a random cubic surface in QpP3 is E#{σf = 0} = (p3 − 1)(p2 + 1) p5 − 1 .
Chiara Meroni Real lines on random cubic surfaces August 28, 2020 23 / 26
Generalizations higher dimension
Setting Every f ∈ R[x0, . . . , xn](d) defines a section σf of the bundle symd(τ ∗
2,n+1):
σf (W ) = f |W . symd(τ ∗
2,n+1)
Gr(2, n + 1)
π σf
When d = 2n − 3 and f is generic, {σf = 0} is a 0-dimensional submanifold, hence there are finitely many lines on Z(f ). Question: What is the expected number of lines on a random hypersurface of degree 2n − 3 in RPn?
Chiara Meroni Real lines on random cubic surfaces August 28, 2020 24 / 26
Generalizations higher dimension
Possible strategy: Redo the proof above in more generality, for an implicit harmonic basis
Chiara Meroni Real lines on random cubic surfaces August 28, 2020 25 / 26
Generalizations higher dimension
In [BLLP] the authors have proved that lim
n→∞
log E Kostlan
n
log Cn = 1 2 Conjecture lim
n→∞
log E TopHarmonic
n
log Cn > 1 2
Chiara Meroni Real lines on random cubic surfaces August 28, 2020 26 / 26