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 Lerario, Erik Lundberg, and Chris Peterson. Random fields and the enumerative geometry of lines on real and complex hypersurfaces. Math. Ann., 374(3-4):1773–1810, 2019. August 28, 2020 2 / 26 Chiara Meroni Real lines on random cubic surfaces

Introduction Setting Let K = R or C and consider f ∈ K [ x 0 , x 1 , x 2 , x 3 ] (3) Definition The cubic surface over the field K associated to f is Z ( f ) = { [ x 0 , x 1 , x 2 , x 3 ] ∈ K P 3 | f ( x 0 , x 1 , x 2 , x 3 ) = 0 } . Z ( f ) ⊂ K P 3 is smooth for the generic choice of f . Question: How many lines are there on a generic cubic surface? August 28, 2020 3 / 26 Chiara Meroni Real lines on random cubic surfaces

Introduction Complex case 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 August 28, 2020 4 / 26 Chiara Meroni Real lines on random cubic surfaces

Introduction Real case 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? August 28, 2020 5 / 26 Chiara Meroni Real lines on random cubic surfaces

Introduction Put a probability distribution on R [ x 0 , x 1 , x 2 , x 3 ] (3) . Requirements: centered gaussian O (4)-invariant classify O (4)-invariant ≡ scalar products classify such distributions on R [ x 0 , x 1 , x 2 , x 3 ] (3) August 28, 2020 6 / 26 Chiara Meroni Real lines on random cubic surfaces

Harmonic decomposition Harmonic decomposition More in general let W n , d = R [ x 0 , . . . x n ] ( d ) ; O ( n + 1) acts on W n , d Aim Find the decomposition of W n , d into its irreducible subrepresentations. Definition Define H n d := { H ∈ W n , d : ∆ H = 0 } to be the space of real homogeneous harmonic polynomials of degree d in n + 1 variables. August 28, 2020 7 / 26 Chiara Meroni Real lines on random cubic surfaces

Harmonic decomposition Decomposition � � x � d − j H n W n , d = j d − j ∈ 2 N Each H n j is O ( n + 1)-invariant and irreducible The decomposition is orthogonal w.r.t. every invariant scalar product j is a multiple of the L 2 scalar Every invariant scalar product on H n product j � x � d − j f j and g = � j � x � d − j g j , Given f , g ∈ W n , d we can write f = � with f j , g j ∈ H n j , and we have that � ( f , g ) = µ j ( f j , g j ) 2 d − j ∈ 2 N for some µ d , µ d − 2 , . . . > 0. August 28, 2020 8 / 26 Chiara Meroni Real lines on random cubic surfaces

Harmonic decomposition From scalar products to probability distributions Fix a harmonic basis { H j , i } ⊂ H n j , orthonormal with respect to ( · , · ) 2 ; then 1 { √ µ j H j , i } is an orthonormal basis of W n , d with respect to ( · , · ). Random polynomial: � � ξ j , i � x � d − j H j , i ( x ) P ( x ) = λ j ξ j , i ∼ N (0 , 1) d − j ∈ 2 N i ∈ J j where J j = dim ( H n j ). August 28, 2020 9 / 26 Chiara Meroni Real lines on random cubic surfaces

Harmonic decomposition Back to our case W 3 , 3 = H 3 3 ⊕ � x � 2 H 3 1 Assume that λ 1 + λ 3 = 1, then     �  + (1 − λ ) � x � 2 � P λ ( x ) = λ ξ 3 , j · H 3 , j ( x ) ξ 1 , j · H 1 , j ( x )  j ∈ J 3 j ∈ J 1 where ξ i , 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]. August 28, 2020 10 / 26 Chiara Meroni Real lines on random cubic surfaces

Kostlan distribution - [BLLP] Kostlan distribution In particular for λ = 1 3 we get the Kostlan polynomial: � 3! � α 0 ! · · · α 3 ! x α 0 0 · · · x α 3 P ( x ) = P 1 3 ( x ) = ξ α · 3 | α | =3 Theorem (Basu, Lerario, Lundberg, Peterson) The expected number E of real lines on a random Kostlan cubic surface in √ R P 3 is E = 6 2 − 3 . August 28, 2020 11 / 26 Chiara Meroni Real lines on random cubic surfaces

Kostlan distribution - [BLLP] Strategy (1) The Grassmannian. Let Gr (2 , 4) denote the Grassmannian of 2-planes in R 4 , and let 2 , 4 ) be the 3 rd symmetric power of the cotangent of the sym 3 ( τ ∗ tautological bundle on Gr (2 , 4). sym 3 ( τ ∗ 2 , 4 ) Every f ∈ R [ x 0 , x 1 , x 2 , x 3 ] (3) defines a section σ f of the bundle sym 3 ( τ ∗ 2 , 4 ): π σ f σ f ( W ) = f | W . Gr (2 , 4) The problem of finding the expected number of lines in the surface Z ( P ) ⊆ R P 3 becomes computing E = E # { W ∈ Gr (2 , 4) | σ P ( W ) = 0 } . August 28, 2020 12 / 26 Chiara Meroni Real lines on random cubic surfaces

Kostlan distribution - [BLLP] (2) Trivialization of the bundle and Kac-Rice formula. Theorem (Kac-Rice formula) Let U ⊂ R N be an open set and X : U → R N be a random map such that: X is gaussian; X is almost surely of class C 1 ; 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 p X ( 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 } p X ( t ) (0) dt B where JX ( t ) denotes the Jacobian matrix of X ( t ) . August 28, 2020 13 / 26 Chiara Meroni Real lines on random cubic surfaces

Kostlan distribution - [BLLP] Kac-Rice formula + trivialization of the (oriented) Grassmannian and its vector bundle ⇓ � E # { ˜ σ P = 0 } = E {| det ( J ( W )) | | ˜ σ P ( W ) = 0 } p (0 , W ) · w Gr + (2 , 4) ( W ) U August 28, 2020 14 / 26 Chiara Meroni Real lines on random cubic surfaces

Kostlan distribution - [BLLP] FACTS: E {| det ( J ( W )) | | ˜ σ P ( W ) = 0 } p (0 , W ) is a constant that does not depend on W , because P is O (4) invariant J ( W ) and ˜ σ P ( W ) are independent E = E # { W ∈ Gr (2 , 4) | σ P λ ( W ) = 0 } = E {| det ( J ( W 0 )) |} · p (0 , W 0 ) · vol ( Gr (2 , 4)) where W 0 = { x 2 = 0 , x 3 = 0 } . August 28, 2020 15 / 26 Chiara Meroni Real lines on random cubic surfaces

Kostlan distribution - [BLLP] (3) E {| det ( J ( W 0 )) |} . Up to a constant the matrix J ( W 0 ) is  0 0  a d √ √ 2 b 2 e a d   ˆ √ √ J =   2 b 2 e  c f    0 c 0 f where a , b , c , d , e , f ∼ N (0 , 1). x = bf − ce | det (ˆ J ) | = | 2 xz − y 2 | y = af − cd � z = ae − bd August 28, 2020 16 / 26 Chiara Meroni Real lines on random cubic surfaces

Kostlan distribution - [BLLP] Finally R 3 | 2 xz − y 2 | e −| ( x , y , z ) | J ) |} = 1 � E {| det (ˆ | ( x , y , z ) | dxdydz 4 π and so √ E = 6 2 − 3 August 28, 2020 17 / 26 Chiara Meroni Real lines on random cubic surfaces

Invariant distributions All invariant distributions Theorem (Ait El Manssour, Belotti, M.) The expected number of real lines on the zero set of the random cubic polynomial P λ equals: E λ = 9(8 λ 2 + (1 − λ ) 2 ) � 8 λ 2 + (1 − λ ) 2 � � 2 λ 2 8 λ 2 + (1 − λ ) 2 − 1 3 + 2 . 2 λ 2 + (1 − λ ) 2 20 λ 2 + (1 − λ ) 2 3 August 28, 2020 18 / 26 Chiara Meroni Real lines on random cubic surfaces

Invariant distributions Strategy (1) The Grassmannian: as the Kostlan case (2) Trivialization of the bundle and Kac-Rice formula: as the Kostlan case so E λ = E {| det ( J ( W 0 )) |} ( λ ) · p (0 , W 0 )( λ ) · vol ( Gr (2 , 4)) (3) E {| det ( J ( W 0 )) |} : the key is to find the correct harmonic basis!     �  + (1 − λ ) � ξ 1 , j · � x � 2 H 1 , j ( x ) P λ ( x ) = λ ξ 3 , j · H 3 , j ( x )  j ∈ J 3 j ∈ J 1 August 28, 2020 19 / 26 Chiara Meroni Real lines on random cubic surfaces

Invariant distributions   a − b 0 d − e 0 a − b d − e c f   J ( W 0 ) =   a + b c d + e f     0 a + b 0 d + e where a ∼ d , b ∼ e , c ∼ f and their variance is a function of λ . E {| det ( J ( W 0 )) |} = � � � λ 2 6 + (1 − λ ) 2 � λ 2 x 2 − λ 4 � λ 2 6 + (1 − λ ) 2 � = 1 � � 4 y 2 + � λ 2 z 2 � · � � 4 π � 48 48 � R 3 � · e −| ( x , y , z ) | | ( x , y , z ) | dxdydz August 28, 2020 20 / 26 Chiara Meroni Real lines on random cubic surfaces

Invariant distributions Function E λ 13 12 11 10 9 8 � 2 Maximum: E 1 = 24 5 − 3 ≃ 12 , 179 7 6 The value λ = 1 corresponds to purely 5 harmonic polynomials of degree 3. 4 3 2 1 0 1 1/3 2/3 August 28, 2020 21 / 26 Chiara Meroni Real lines on random cubic surfaces