How to Park your Car on a Moduli Space Maria Gillespie, Colorado State University On joint work with Renzo Cavalieri (CSU) and Leonid Monin (University of Toronto) University of Colorado Boulder, Feb 11, 2020 M. Gillespie How to Park your Car on a Moduli Space Feb 11, 2020 1 / 15
Background: Enumerative geometry Q1: Given four lines ℓ 1 , ℓ 2 , ℓ 3 , ℓ 4 in ‘general position’ in three-dimensional space, how many lines pass through all four of them? M. Gillespie How to Park your Car on a Moduli Space Feb 11, 2020 2 / 15
Background: Enumerative geometry Q1: Given four lines ℓ 1 , ℓ 2 , ℓ 3 , ℓ 4 in ‘general position’ in three-dimensional space, how many lines pass through all four of them? A1: At most 2 (over R ). A2: Exactly 2 (over C , in projective space). M. Gillespie How to Park your Car on a Moduli Space Feb 11, 2020 2 / 15
Background: Enumerative geometry Q2: How many lines lie on a generic cubic surface in CP 3 ? A: 27 M. Gillespie How to Park your Car on a Moduli Space Feb 11, 2020 3 / 15
Solving Enumerative Geometry Problems in Three Easy Steps! M. Gillespie How to Park your Car on a Moduli Space Feb 11, 2020 4 / 15
Solving Enumerative Geometry Problems in Three Easy Steps! 1 Make the problem more difficult. M. Gillespie How to Park your Car on a Moduli Space Feb 11, 2020 4 / 15
Solving Enumerative Geometry Problems in Three Easy Steps! 1 Make the problem more difficult. 2 Make it even harder. M. Gillespie How to Park your Car on a Moduli Space Feb 11, 2020 4 / 15
Solving Enumerative Geometry Problems in Three Easy Steps! 1 Make the problem more difficult. 2 Make it even harder. 3 Turn it into a combinatorics problem, which may or may not be easier to solve than the original geometry question. M. Gillespie How to Park your Car on a Moduli Space Feb 11, 2020 4 / 15
Solving Enumerative Geometry Problems in Three Easy Steps! 1 Make the problem more difficult. 2 Make it even harder. 3 Turn it into a combinatorics problem, which may or may not be easier to solve than the original geometry question. Fine Print: 1 Rephrase the Geometry: Turn it into an intersection problem about families of geometric objects in some larger ’moduli space’. 2 Geometry Ñ Algebra: State the intersection problem algebraically (in terms of the ‘cohomology ring’ or ‘chow ring’ of the moduli space). 3 Algebra Ñ Combinatorics: Determine the combinatorial structure of the algebraic space, and use it to phrase the intersection problem in terms of simple combinatorial objects. Solve. M. Gillespie How to Park your Car on a Moduli Space Feb 11, 2020 4 / 15
Example of the Three-Step Method Q1: Given four lines ℓ 1 , ℓ 2 , ℓ 3 , ℓ 4 in ‘general position’ in CP 3 , how many lines pass through all four of them? M. Gillespie How to Park your Car on a Moduli Space Feb 11, 2020 5 / 15
Example of the Three-Step Method Q1: Given four lines ℓ 1 , ℓ 2 , ℓ 3 , ℓ 4 in ‘general position’ in CP 3 , how many lines pass through all four of them? 1 Rephrase the Geometry: § X i : “ t lines passing through ℓ i u . § X i Ď Gr , where Gr is the moduli space of all lines in CP 3 . § Want to compute | X 1 X X 2 X X 3 X X 4 | . M. Gillespie How to Park your Car on a Moduli Space Feb 11, 2020 5 / 15
Example of the Three-Step Method Q1: Given four lines ℓ 1 , ℓ 2 , ℓ 3 , ℓ 4 in ‘general position’ in CP 3 , how many lines pass through all four of them? 1 Rephrase the Geometry: § X i : “ t lines passing through ℓ i u . § X i Ď Gr , where Gr is the moduli space of all lines in CP 3 . § Want to compute | X 1 X X 2 X X 3 X X 4 | . 2 Geometry Ñ Algebra: § r X 1 s “ r X 2 s “ r X 3 s “ r X 4 s P H ˚ p Gr q , multiplication in H ˚ corresponds to intersection (for generic representatives of each class). § Want to compute r X 1 s 4 . M. Gillespie How to Park your Car on a Moduli Space Feb 11, 2020 5 / 15
Example of the Three-Step Method Q1: Given four lines ℓ 1 , ℓ 2 , ℓ 3 , ℓ 4 in ‘general position’ in CP 3 , how many lines pass through all four of them? 1 Rephrase the Geometry: § X i : “ t lines passing through ℓ i u . § X i Ď Gr , where Gr is the moduli space of all lines in CP 3 . § Want to compute | X 1 X X 2 X X 3 X X 4 | . 2 Geometry Ñ Algebra: § r X 1 s “ r X 2 s “ r X 3 s “ r X 4 s P H ˚ p Gr q , multiplication in H ˚ corresponds to intersection (for generic representatives of each class). § Want to compute r X 1 s 4 . 3 Algebra Ñ Combinatorics: § H ˚ p Gr q is isomorphic to a quotient of the ring of symmetric functions § r X 1 s 4 “ c ¨ r pt s where c “ # ways to fill a 2 ˆ 2 grid with 1 , 2 , 3 , 4 with increasing rows and columns: 2 4 3 4 1 3 1 2 M. Gillespie How to Park your Car on a Moduli Space Feb 11, 2020 5 / 15
Moduli space of genus-0 curves M 0 , n is space of (isom. classes of) complex ‘stable curves’ of genus 0 with n distinct ‘marked points’. Example: CP 1 , need to mark at least 3 points for it to be ‘stable’: no nontrivial automorphisms. M. Gillespie How to Park your Car on a Moduli Space Feb 11, 2020 6 / 15
Moduli space of genus-0 curves M 0 , n is space of (isom. classes of) complex ‘stable curves’ of genus 0 with n distinct ‘marked points’. Example: CP 1 , need to mark at least 3 points for it to be ‘stable’: no nontrivial automorphisms. In general: Allow several copies of CP 1 glued together at nodes, where each sphere has a total of at least 3 nodes and marked points. M. Gillespie How to Park your Car on a Moduli Space Feb 11, 2020 6 / 15
How do you draw M 0 , n ? | M 0 , 3 | “ 1 - only one stable curve with 3 points up to isomorphism M 0 , 4 ” P 1 : fix 3 points, let the fourth vary. M. Gillespie How to Park your Car on a Moduli Space Feb 11, 2020 7 / 15
How do you draw M 0 , n ? | M 0 , 3 | “ 1 - only one stable curve with 3 points up to isomorphism M 0 , 4 ” P 1 : fix 3 points, let the fourth vary. In general dim M 0 , n ` 3 “ n , but not isomorphic to P n . Kapranov map : M 0 , n ` 3 Ñ P n M. Gillespie How to Park your Car on a Moduli Space Feb 11, 2020 7 / 15
How do you draw M 0 , n ? | M 0 , 3 | “ 1 - only one stable curve with 3 points up to isomorphism M 0 , 4 ” P 1 : fix 3 points, let the fourth vary. In general dim M 0 , n ` 3 “ n , but not isomorphic to P n . Kapranov map : M 0 , n ` 3 Ñ P n Forgetting map : M 0 , n ` 3 Ñ M 0 , n ` 2 by forgetting last marked point (and stabilizing) (Keel, Tevelev) : Combining Kapranov, forgetting gives embedding Ñ M 0 , n ` 2 ˆ P n M 0 , n ` 3 ã M. Gillespie How to Park your Car on a Moduli Space Feb 11, 2020 7 / 15
How do you draw M 0 , n ? | M 0 , 3 | “ 1 - only one stable curve with 3 points up to isomorphism M 0 , 4 ” P 1 : fix 3 points, let the fourth vary. In general dim M 0 , n ` 3 “ n , but not isomorphic to P n . Kapranov map : M 0 , n ` 3 Ñ P n Forgetting map : M 0 , n ` 3 Ñ M 0 , n ` 2 by forgetting last marked point (and stabilizing) (Keel, Tevelev) : Combining Kapranov, forgetting gives embedding Ñ M 0 , n ` 2 ˆ P n M 0 , n ` 3 ã Iterate: Get embedding Ñ P 1 ˆ P 2 ˆ ¨ ¨ ¨ ˆ P n M 0 , n ` 3 ã M. Gillespie How to Park your Car on a Moduli Space Feb 11, 2020 7 / 15
Degree of an embedding: How ugly is your drawing? Ñ P n and dim p X q “ d . Then the degree of this Suppose X ã embedding is the number of points in an intersection of X with d generic hyperplanes in P n . M. Gillespie How to Park your Car on a Moduli Space Feb 11, 2020 8 / 15
Degree of an embedding: How ugly is your drawing? Ñ P n and dim p X q “ d . Then the degree of this Suppose X ã embedding is the number of points in an intersection of X with d generic hyperplanes in P n . M. Gillespie How to Park your Car on a Moduli Space Feb 11, 2020 8 / 15
Degree of an embedding: How ugly is your drawing? Ñ P n and dim p X q “ d . Then the degree of this Suppose X ã embedding is the number of points in an intersection of X with d generic hyperplanes in P n . M. Gillespie How to Park your Car on a Moduli Space Feb 11, 2020 8 / 15
Degree of an embedding: How ugly is your drawing? Ñ P n and dim p X q “ d . Then the degree of this Suppose X ã embedding is the number of points in an intersection of X with d generic hyperplanes in P n . M. Gillespie How to Park your Car on a Moduli Space Feb 11, 2020 8 / 15
Multidegree: How ugly is your ‘multi-drawing’? Ñ P 1 ˆ ¨ ¨ ¨ ˆ P n . Then the p k 1 , . . . , k n q - multidegree , Suppose X ã deg k 1 ,..., k n p X q , is the expected size of intersection of X with a total of k 1 ` ¨ ¨ ¨ ` k n hyperplanes, k i of which are from P i for each i , where k 1 ` ¨ ¨ ¨ ` k n “ d . M. Gillespie How to Park your Car on a Moduli Space Feb 11, 2020 9 / 15
Multidegree: How ugly is your ‘multi-drawing’? Ñ P 1 ˆ ¨ ¨ ¨ ˆ P n . Then the p k 1 , . . . , k n q - multidegree , Suppose X ã deg k 1 ,..., k n p X q , is the expected size of intersection of X with a total of k 1 ` ¨ ¨ ¨ ` k n hyperplanes, k i of which are from P i for each i , where k 1 ` ¨ ¨ ¨ ` k n “ d . Total degree (Van der Waarden:) Let C be projectivization of preimage of X in affine space A 2 ˆ ¨ ¨ ¨ ˆ A n ` 1 “ A p n ` 1 qp n ` 2 q{ 2 ´ 1 . Then ÿ deg p C q “ deg k 1 ,..., k n X . p k 1 ,..., k n q M. Gillespie How to Park your Car on a Moduli Space Feb 11, 2020 9 / 15
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