enumerative geometry and the shapiro shapiro...
TRANSCRIPT
![Page 1: Enumerative Geometry and the Shapiro Shapiro Conjecturesites.math.washington.edu/~jlev/sfu_slides.pdf · Enumerative geometry Some questions about 3D geometry: 1.How many lines meet](https://reader035.vdocument.in/reader035/viewer/2022080717/5f7815655446ce6dd9227922/html5/thumbnails/1.jpg)
Enumerative Geometryand the Shapiro–Shapiro Conjecture
Jake LevinsonUniversity of Washington
Simon Fraser UniversityNovember 20, 2019
![Page 2: Enumerative Geometry and the Shapiro Shapiro Conjecturesites.math.washington.edu/~jlev/sfu_slides.pdf · Enumerative geometry Some questions about 3D geometry: 1.How many lines meet](https://reader035.vdocument.in/reader035/viewer/2022080717/5f7815655446ce6dd9227922/html5/thumbnails/2.jpg)
Enumerative geometry
Some questions about 3D geometry:
1. How many lines meet four general lines?
2.
2. How many lines lie on a smooth cubic surface?
27.
3. How many rational cubic curves meet 12 lines?
80160.
![Page 3: Enumerative Geometry and the Shapiro Shapiro Conjecturesites.math.washington.edu/~jlev/sfu_slides.pdf · Enumerative geometry Some questions about 3D geometry: 1.How many lines meet](https://reader035.vdocument.in/reader035/viewer/2022080717/5f7815655446ce6dd9227922/html5/thumbnails/3.jpg)
Enumerative geometry
Some questions about 3D geometry:
1. How many lines meet four general lines?
2.
2. How many lines lie on a smooth cubic surface?
27.
3. How many rational cubic curves meet 12 lines?
80160.
![Page 4: Enumerative Geometry and the Shapiro Shapiro Conjecturesites.math.washington.edu/~jlev/sfu_slides.pdf · Enumerative geometry Some questions about 3D geometry: 1.How many lines meet](https://reader035.vdocument.in/reader035/viewer/2022080717/5f7815655446ce6dd9227922/html5/thumbnails/4.jpg)
Enumerative geometry
Some questions about 3D geometry:
1. How many lines meet four general lines? 2.
2. How many lines lie on a smooth cubic surface?
27.
3. How many rational cubic curves meet 12 lines?
80160.
![Page 5: Enumerative Geometry and the Shapiro Shapiro Conjecturesites.math.washington.edu/~jlev/sfu_slides.pdf · Enumerative geometry Some questions about 3D geometry: 1.How many lines meet](https://reader035.vdocument.in/reader035/viewer/2022080717/5f7815655446ce6dd9227922/html5/thumbnails/5.jpg)
Enumerative geometry
Some questions about 3D geometry:
1. How many lines meet four general lines? 2.
2. How many lines lie on a smooth cubic surface?
27.
3. How many rational cubic curves meet 12 lines?
80160.
![Page 6: Enumerative Geometry and the Shapiro Shapiro Conjecturesites.math.washington.edu/~jlev/sfu_slides.pdf · Enumerative geometry Some questions about 3D geometry: 1.How many lines meet](https://reader035.vdocument.in/reader035/viewer/2022080717/5f7815655446ce6dd9227922/html5/thumbnails/6.jpg)
Enumerative geometry
Some questions about 3D geometry:
1. How many lines meet four general lines? 2.
2. How many lines lie on a smooth cubic surface? 27.
3. How many rational cubic curves meet 12 lines?
80160.
(Credit: Greg Egan)
![Page 7: Enumerative Geometry and the Shapiro Shapiro Conjecturesites.math.washington.edu/~jlev/sfu_slides.pdf · Enumerative geometry Some questions about 3D geometry: 1.How many lines meet](https://reader035.vdocument.in/reader035/viewer/2022080717/5f7815655446ce6dd9227922/html5/thumbnails/7.jpg)
Enumerative geometry
Some questions about 3D geometry:
1. How many lines meet four general lines? 2.
2. How many lines lie on a smooth cubic surface? 27.
3. How many rational cubic curves meet 12 lines?
80160.
![Page 8: Enumerative Geometry and the Shapiro Shapiro Conjecturesites.math.washington.edu/~jlev/sfu_slides.pdf · Enumerative geometry Some questions about 3D geometry: 1.How many lines meet](https://reader035.vdocument.in/reader035/viewer/2022080717/5f7815655446ce6dd9227922/html5/thumbnails/8.jpg)
Enumerative geometry
Some questions about 3D geometry:
1. How many lines meet four general lines? 2.
2. How many lines lie on a smooth cubic surface? 27.
3. How many rational cubic curves meet 12 lines? 80160.
![Page 9: Enumerative Geometry and the Shapiro Shapiro Conjecturesites.math.washington.edu/~jlev/sfu_slides.pdf · Enumerative geometry Some questions about 3D geometry: 1.How many lines meet](https://reader035.vdocument.in/reader035/viewer/2022080717/5f7815655446ce6dd9227922/html5/thumbnails/9.jpg)
Enumerative geometry
Some questions about 3D geometry:
1. How many lines meet four general lines? 2.
2. How many lines lie on a smooth cubic surface? 27.
3. How many rational cubic curves meet 12 lines? 80160.
Prettier than 80160 twisted cubics.
![Page 10: Enumerative Geometry and the Shapiro Shapiro Conjecturesites.math.washington.edu/~jlev/sfu_slides.pdf · Enumerative geometry Some questions about 3D geometry: 1.How many lines meet](https://reader035.vdocument.in/reader035/viewer/2022080717/5f7815655446ce6dd9227922/html5/thumbnails/10.jpg)
Enumerative geometry
Some questions about 3D geometry:
1. How many lines meet four general lines? 2.
2. How many lines lie on a smooth cubic surface? 27.
3. How many rational cubic curves meet 12 lines? 80160.
![Page 11: Enumerative Geometry and the Shapiro Shapiro Conjecturesites.math.washington.edu/~jlev/sfu_slides.pdf · Enumerative geometry Some questions about 3D geometry: 1.How many lines meet](https://reader035.vdocument.in/reader035/viewer/2022080717/5f7815655446ce6dd9227922/html5/thumbnails/11.jpg)
Enumerative geometry
Some questions about 3D geometry:
1. How many lines meet four general lines? 2.
2. How many lines lie on a smooth cubic surface? 27.
3. How many rational cubic curves meet 12 lines? 80160.
To answer these questions, we study moduli spaces:
I The space of lines Gr(2, n)
I The space of curves Mg
I The space of maps of curves M0,n(P3, 3)
![Page 12: Enumerative Geometry and the Shapiro Shapiro Conjecturesites.math.washington.edu/~jlev/sfu_slides.pdf · Enumerative geometry Some questions about 3D geometry: 1.How many lines meet](https://reader035.vdocument.in/reader035/viewer/2022080717/5f7815655446ce6dd9227922/html5/thumbnails/12.jpg)
Enumerative geometry
Some questions about 3D geometry:
1. How many lines meet four general lines? 2.
2. How many lines lie on a smooth cubic surface? 27.
3. How many rational cubic curves meet 12 lines? 80160.
To answer these questions, we study moduli spaces:
I The space of lines Gr(2, n)
I The space of curves Mg
I The space of maps of curves M0,n(P3, 3)
Hilbert’s 15th Problem (1900): “To establish rigorously ... theenumerative calculus developed by [Schubert].”
Mumford (1983): “We take as a model for [enumeration onMg ]the enumerative geometry of the Grassmannians.”
![Page 13: Enumerative Geometry and the Shapiro Shapiro Conjecturesites.math.washington.edu/~jlev/sfu_slides.pdf · Enumerative geometry Some questions about 3D geometry: 1.How many lines meet](https://reader035.vdocument.in/reader035/viewer/2022080717/5f7815655446ce6dd9227922/html5/thumbnails/13.jpg)
Modern questions also go beyond enumeration
I Euler characteristic, genus, . . . for positive-dimensionalsolution spaces
I Equivariant K-theory of Grassmannians. O. Pechenik and A. Yong.Forum Math. Pi (2017).
I Explicit topology of moduli spaces (e.g. as CW-complexes)I Combinatorial equivalence of real moduli spaces. S. Devadoss.
Notices Amer. Math. Soc. (2004).
I Deforming solutionsI Genera of Brill-Noether curves and staircase paths in Young
tableaux, M. Chan; A. Lopez Martın; N. Pflueger; M. Teixidor iBigas. Trans. Amer. Math. Soc. (2018).
Goal of today
Pass from counting to describing topology.
![Page 14: Enumerative Geometry and the Shapiro Shapiro Conjecturesites.math.washington.edu/~jlev/sfu_slides.pdf · Enumerative geometry Some questions about 3D geometry: 1.How many lines meet](https://reader035.vdocument.in/reader035/viewer/2022080717/5f7815655446ce6dd9227922/html5/thumbnails/14.jpg)
Modern questions also go beyond enumeration
I Euler characteristic, genus, . . . for positive-dimensionalsolution spaces
I Equivariant K-theory of Grassmannians. O. Pechenik and A. Yong.Forum Math. Pi (2017).
I Explicit topology of moduli spaces (e.g. as CW-complexes)I Combinatorial equivalence of real moduli spaces. S. Devadoss.
Notices Amer. Math. Soc. (2004).
I Deforming solutionsI Genera of Brill-Noether curves and staircase paths in Young
tableaux, M. Chan; A. Lopez Martın; N. Pflueger; M. Teixidor iBigas. Trans. Amer. Math. Soc. (2018).
Goal of today
Pass from counting to describing topology.
![Page 15: Enumerative Geometry and the Shapiro Shapiro Conjecturesites.math.washington.edu/~jlev/sfu_slides.pdf · Enumerative geometry Some questions about 3D geometry: 1.How many lines meet](https://reader035.vdocument.in/reader035/viewer/2022080717/5f7815655446ce6dd9227922/html5/thumbnails/15.jpg)
Part 1. Counting
![Page 16: Enumerative Geometry and the Shapiro Shapiro Conjecturesites.math.washington.edu/~jlev/sfu_slides.pdf · Enumerative geometry Some questions about 3D geometry: 1.How many lines meet](https://reader035.vdocument.in/reader035/viewer/2022080717/5f7815655446ce6dd9227922/html5/thumbnails/16.jpg)
Counting lines and planes
How many lines meet four general lines?
Schubert calculus says: 2 solutions – enumerated by the set{1 23 4 , 1 3
2 4
}.
Caveat: in general, over C (in CP3, with multiplicity, ...).
![Page 17: Enumerative Geometry and the Shapiro Shapiro Conjecturesites.math.washington.edu/~jlev/sfu_slides.pdf · Enumerative geometry Some questions about 3D geometry: 1.How many lines meet](https://reader035.vdocument.in/reader035/viewer/2022080717/5f7815655446ce6dd9227922/html5/thumbnails/17.jpg)
Counting lines and planes
How many lines meet four general lines?
Schubert calculus says: 2 solutions – enumerated by the set{1 23 4 , 1 3
2 4
}.
Caveat: in general, over C (in CP3, with multiplicity, ...).
![Page 18: Enumerative Geometry and the Shapiro Shapiro Conjecturesites.math.washington.edu/~jlev/sfu_slides.pdf · Enumerative geometry Some questions about 3D geometry: 1.How many lines meet](https://reader035.vdocument.in/reader035/viewer/2022080717/5f7815655446ce6dd9227922/html5/thumbnails/18.jpg)
Schubert problems, k-planes, flags
The Grassmannian is the space of planes:
Gr(k , n) = {vector subspaces S ⊂ Cn : dim(S) = k}.
Simplest (codimension 1) “Schubert problem” for planes:
X (Fn−k) = {S ∈ Gr(k , n) : S ∩ Fn−k 6= 0}.
In P3: Lines meeting a given line.
Theorem. For general choices of complementary planes F (i),
Zk,n := X (F(1)n−k) ∩ · · · ∩ X (F
(k(n−k))n−k ) is finite
and counted by standard Young tableaux:
#Zk,n = #SYT ( ) =
{1 2 43 5 6 ,
1 3 52 4 6 , · · ·
}.
![Page 19: Enumerative Geometry and the Shapiro Shapiro Conjecturesites.math.washington.edu/~jlev/sfu_slides.pdf · Enumerative geometry Some questions about 3D geometry: 1.How many lines meet](https://reader035.vdocument.in/reader035/viewer/2022080717/5f7815655446ce6dd9227922/html5/thumbnails/19.jpg)
Schubert problems, k-planes, flags
The Grassmannian is the space of planes:
Gr(k , n) = {vector subspaces S ⊂ Cn : dim(S) = k}.
Simplest (codimension 1) “Schubert problem” for planes:
X (Fn−k) = {S ∈ Gr(k , n) : S ∩ Fn−k 6= 0}.
In P3: Lines meeting a given line.
Theorem. For general choices of complementary planes F (i),
Zk,n := X (F(1)n−k) ∩ · · · ∩ X (F
(k(n−k))n−k ) is finite
and counted by standard Young tableaux:
#Zk,n = #SYT ( ) =
{1 2 43 5 6 ,
1 3 52 4 6 , · · ·
}.
![Page 20: Enumerative Geometry and the Shapiro Shapiro Conjecturesites.math.washington.edu/~jlev/sfu_slides.pdf · Enumerative geometry Some questions about 3D geometry: 1.How many lines meet](https://reader035.vdocument.in/reader035/viewer/2022080717/5f7815655446ce6dd9227922/html5/thumbnails/20.jpg)
Schubert problems, k-planes, flags
The Grassmannian is the space of planes:
Gr(k , n) = {vector subspaces S ⊂ Cn : dim(S) = k}.
Simplest (codimension 1) “Schubert problem” for planes:
X (Fn−k) = {S ∈ Gr(k , n) : S ∩ Fn−k 6= 0}.
In P3: Lines meeting a given line.
Theorem. For general choices of complementary planes F (i),
Zk,n := X (F(1)n−k) ∩ · · · ∩ X (F
(k(n−k))n−k ) is finite
and counted by standard Young tableaux:
#Zk,n = #SYT ( ) =
{1 2 43 5 6 ,
1 3 52 4 6 , · · ·
}.
![Page 21: Enumerative Geometry and the Shapiro Shapiro Conjecturesites.math.washington.edu/~jlev/sfu_slides.pdf · Enumerative geometry Some questions about 3D geometry: 1.How many lines meet](https://reader035.vdocument.in/reader035/viewer/2022080717/5f7815655446ce6dd9227922/html5/thumbnails/21.jpg)
Schubert problems, k-planes, flags
The Grassmannian is the space of planes:
Gr(k , n) = {vector subspaces S ⊂ Cn : dim(S) = k}.
Simplest (codimension 1) “Schubert problem” for planes:
X (Fn−k) = {S ∈ Gr(k , n) : S ∩ Fn−k 6= 0}.
In P3: Lines meeting a given line.
Theorem. For general choices of complementary planes F (i),
Zk,n := X (F(1)n−k) ∩ · · · ∩ X (F
(k(n−k))n−k ) is finite
and counted by standard Young tableaux:
#Zk,n = #SYT ( ) =
{1 2 43 5 6 ,
1 3 52 4 6 , · · ·
}.
![Page 22: Enumerative Geometry and the Shapiro Shapiro Conjecturesites.math.washington.edu/~jlev/sfu_slides.pdf · Enumerative geometry Some questions about 3D geometry: 1.How many lines meet](https://reader035.vdocument.in/reader035/viewer/2022080717/5f7815655446ce6dd9227922/html5/thumbnails/22.jpg)
Choosing flags
Theorem. For general choices of F (i),
Zk,n = X (F(1)n−k) ∩ · · · ∩ X (F
(k(n−k))n−k )
is finite and counted by standard Young tableaux:
#Zk,n = #SYT( ) =
{1 2 43 5 6 ,
1 3 52 4 6 , · · ·
}.
General Schubert problems: consider S ∩F , for a complete flag:
F : F1 ⊂ F2 ⊂ · · · ⊂ Fn = Cn.
Problems:
1. In bad cases, Zk,n might have multiplicity (or be infinite).
2. No canonical bijection Zk,n ↔ SYT( ) in general.
![Page 23: Enumerative Geometry and the Shapiro Shapiro Conjecturesites.math.washington.edu/~jlev/sfu_slides.pdf · Enumerative geometry Some questions about 3D geometry: 1.How many lines meet](https://reader035.vdocument.in/reader035/viewer/2022080717/5f7815655446ce6dd9227922/html5/thumbnails/23.jpg)
Choosing flags
Theorem. For general choices of F (i),
Zk,n = X (F(1)n−k) ∩ · · · ∩ X (F
(k(n−k))n−k )
is finite and counted by standard Young tableaux:
#Zk,n = #SYT( ) =
{1 2 43 5 6 ,
1 3 52 4 6 , · · ·
}.
General Schubert problems: consider S ∩F , for a complete flag:
F : F1 ⊂ F2 ⊂ · · · ⊂ Fn = Cn.
Problems:
1. In bad cases, Zk,n might have multiplicity (or be infinite).
2. No canonical bijection Zk,n ↔ SYT( ) in general.
![Page 24: Enumerative Geometry and the Shapiro Shapiro Conjecturesites.math.washington.edu/~jlev/sfu_slides.pdf · Enumerative geometry Some questions about 3D geometry: 1.How many lines meet](https://reader035.vdocument.in/reader035/viewer/2022080717/5f7815655446ce6dd9227922/html5/thumbnails/24.jpg)
Choosing flags
Theorem. For general choices of F (i),
Zk,n = X (F(1)n−k) ∩ · · · ∩ X (F
(k(n−k))n−k )
is finite and counted by standard Young tableaux:
#Zk,n = #SYT( ) =
{1 2 43 5 6 ,
1 3 52 4 6 , · · ·
}.
General Schubert problems: consider S ∩F , for a complete flag:
F : F1 ⊂ F2 ⊂ · · · ⊂ Fn = Cn.
Problems:
1. In bad cases, Zk,n might have multiplicity (or be infinite).
2. No canonical bijection Zk,n ↔ SYT( ) in general.
![Page 25: Enumerative Geometry and the Shapiro Shapiro Conjecturesites.math.washington.edu/~jlev/sfu_slides.pdf · Enumerative geometry Some questions about 3D geometry: 1.How many lines meet](https://reader035.vdocument.in/reader035/viewer/2022080717/5f7815655446ce6dd9227922/html5/thumbnails/25.jpg)
Tangent flags to the rational normal curve
I The rational normal curve in Pn−1:
P1 ↪→ P(Cn) = Pn−1 by
t 7→ [1 : t : t2 : · · · : tn−1]
I (Maximally) tangent flag F (t), t ∈ P1:
I Schubert problems using F (t) relate to moduli of curves:I (limit) linear series, Weierstrass points, Brill–Noether loci, . . .
I Eisenbud–Harris ’83:Solutions to these problems have the expected dimension.
![Page 26: Enumerative Geometry and the Shapiro Shapiro Conjecturesites.math.washington.edu/~jlev/sfu_slides.pdf · Enumerative geometry Some questions about 3D geometry: 1.How many lines meet](https://reader035.vdocument.in/reader035/viewer/2022080717/5f7815655446ce6dd9227922/html5/thumbnails/26.jpg)
Tangent flags to the rational normal curve
I The rational normal curve in Pn−1:
P1 ↪→ P(Cn) = Pn−1 by
t 7→ [1 : t : t2 : · · · : tn−1]
I (Maximally) tangent flag F (t), t ∈ P1:
I Schubert problems using F (t) relate to moduli of curves:I (limit) linear series, Weierstrass points, Brill–Noether loci, . . .
I Eisenbud–Harris ’83:Solutions to these problems have the expected dimension.
![Page 27: Enumerative Geometry and the Shapiro Shapiro Conjecturesites.math.washington.edu/~jlev/sfu_slides.pdf · Enumerative geometry Some questions about 3D geometry: 1.How many lines meet](https://reader035.vdocument.in/reader035/viewer/2022080717/5f7815655446ce6dd9227922/html5/thumbnails/27.jpg)
Tangent flags to the rational normal curve
I The rational normal curve in Pn−1:
P1 ↪→ P(Cn) = Pn−1 by
t 7→ [1 : t : t2 : · · · : tn−1]
I (Maximally) tangent flag F (t), t ∈ P1:
I Schubert problems using F (t) relate to moduli of curves:I (limit) linear series, Weierstrass points, Brill–Noether loci, . . .
I Eisenbud–Harris ’83:Solutions to these problems have the expected dimension.
![Page 28: Enumerative Geometry and the Shapiro Shapiro Conjecturesites.math.washington.edu/~jlev/sfu_slides.pdf · Enumerative geometry Some questions about 3D geometry: 1.How many lines meet](https://reader035.vdocument.in/reader035/viewer/2022080717/5f7815655446ce6dd9227922/html5/thumbnails/28.jpg)
Tangent flags to the rational normal curve
I The rational normal curve in Pn−1:
P1 ↪→ P(Cn) = Pn−1 by
t 7→ [1 : t : t2 : · · · : tn−1]
I (Maximally) tangent flag F (t), t ∈ P1:
I Schubert problems using F (t) relate to moduli of curves:I (limit) linear series, Weierstrass points, Brill–Noether loci, . . .
I Eisenbud–Harris ’83:Solutions to these problems have the expected dimension.
![Page 29: Enumerative Geometry and the Shapiro Shapiro Conjecturesites.math.washington.edu/~jlev/sfu_slides.pdf · Enumerative geometry Some questions about 3D geometry: 1.How many lines meet](https://reader035.vdocument.in/reader035/viewer/2022080717/5f7815655446ce6dd9227922/html5/thumbnails/29.jpg)
Schubert calculus over R?
Conjecture (Shapiro–Shapiro ’95)
For any choice of distinct real t1, . . . , tN ∈ RP1,
Zk,n = X (F (t1)n−k) ∩ · · · ∩ X (F (tN)n−k)
consists entirely of real, multiplicity-free points.
Proven in 2005 / 2009:I Schubert calculus and representations of the general linear group.
Mukhin, E.; Tarasov, V.; Varchenko, A., J. Amer. Math. Soc. (2009).
I The B. and M. Shapiro conjecture in real algebraic geometry and theBethe ansatz. Mukhin, E; Tarasov, V.; Varchenko, A., Ann. of Math.(2009).
Case k = 2 established earlier:I Rational functions with real critical points and the B. and M. Shapiro
conjecture in real enumerative geometry. Eremenko, A. and Gabrielov, A.Ann. of Math. (2002).
![Page 30: Enumerative Geometry and the Shapiro Shapiro Conjecturesites.math.washington.edu/~jlev/sfu_slides.pdf · Enumerative geometry Some questions about 3D geometry: 1.How many lines meet](https://reader035.vdocument.in/reader035/viewer/2022080717/5f7815655446ce6dd9227922/html5/thumbnails/30.jpg)
Schubert calculus over R?
Conjecture (Shapiro–Shapiro ’95)
For any choice of distinct real t1, . . . , tN ∈ RP1,
Zk,n = X (F (t1)n−k) ∩ · · · ∩ X (F (tN)n−k)
consists entirely of real, multiplicity-free points.
Proven in 2005 / 2009:I Schubert calculus and representations of the general linear group.
Mukhin, E.; Tarasov, V.; Varchenko, A., J. Amer. Math. Soc. (2009).
I The B. and M. Shapiro conjecture in real algebraic geometry and theBethe ansatz. Mukhin, E; Tarasov, V.; Varchenko, A., Ann. of Math.(2009).
Case k = 2 established earlier:I Rational functions with real critical points and the B. and M. Shapiro
conjecture in real enumerative geometry. Eremenko, A. and Gabrielov, A.Ann. of Math. (2002).
![Page 31: Enumerative Geometry and the Shapiro Shapiro Conjecturesites.math.washington.edu/~jlev/sfu_slides.pdf · Enumerative geometry Some questions about 3D geometry: 1.How many lines meet](https://reader035.vdocument.in/reader035/viewer/2022080717/5f7815655446ce6dd9227922/html5/thumbnails/31.jpg)
Schubert calculus over R?
Theorem (M–T–V ’05)
For any choice of distinct real t1, . . . , tN ∈ RP1,
Zk,n = X (F (t1)n−k) ∩ · · · ∩ X (F (tN)n−k)
consists entirely of real, multiplicity-free points.
Proven in 2005 / 2009:I Schubert calculus and representations of the general linear group.
Mukhin, E.; Tarasov, V.; Varchenko, A., J. Amer. Math. Soc. (2009).
I The B. and M. Shapiro conjecture in real algebraic geometry and theBethe ansatz. Mukhin, E; Tarasov, V.; Varchenko, A., Ann. of Math.(2009).
Case k = 2 established earlier:I Rational functions with real critical points and the B. and M. Shapiro
conjecture in real enumerative geometry. Eremenko, A. and Gabrielov, A.Ann. of Math. (2002).
![Page 32: Enumerative Geometry and the Shapiro Shapiro Conjecturesites.math.washington.edu/~jlev/sfu_slides.pdf · Enumerative geometry Some questions about 3D geometry: 1.How many lines meet](https://reader035.vdocument.in/reader035/viewer/2022080717/5f7815655446ce6dd9227922/html5/thumbnails/32.jpg)
Combinatorial consequences
Theorem (M–T–V ’05)
For any choice of distinct real t1, . . . , tN ∈ RP1,
Zk,n = X (F (t1)n−k) ∩ · · · ∩ X (F (tN)n−k)
consists entirely of real, multiplicity-free points.
I The cardinality of Zk,n is always exactly #SYT.
I This suggests there may be a canonical bijection
Zk,n ↔ SYT.
![Page 33: Enumerative Geometry and the Shapiro Shapiro Conjecturesites.math.washington.edu/~jlev/sfu_slides.pdf · Enumerative geometry Some questions about 3D geometry: 1.How many lines meet](https://reader035.vdocument.in/reader035/viewer/2022080717/5f7815655446ce6dd9227922/html5/thumbnails/33.jpg)
Combinatorial consequences
Theorem (M–T–V ’05)
For any choice of distinct real t1, . . . , tN ∈ RP1,
Zk,n = X (F (t1)n−k) ∩ · · · ∩ X (F (tN)n−k)
consists entirely of real, multiplicity-free points.
I The cardinality of Zk,n is always exactly #SYT.
I This suggests there may be a canonical bijection
Zk,n ↔ SYT.
![Page 34: Enumerative Geometry and the Shapiro Shapiro Conjecturesites.math.washington.edu/~jlev/sfu_slides.pdf · Enumerative geometry Some questions about 3D geometry: 1.How many lines meet](https://reader035.vdocument.in/reader035/viewer/2022080717/5f7815655446ce6dd9227922/html5/thumbnails/34.jpg)
Part 2. Labeling
![Page 35: Enumerative Geometry and the Shapiro Shapiro Conjecturesites.math.washington.edu/~jlev/sfu_slides.pdf · Enumerative geometry Some questions about 3D geometry: 1.How many lines meet](https://reader035.vdocument.in/reader035/viewer/2022080717/5f7815655446ce6dd9227922/html5/thumbnails/35.jpg)
Families of geometry problems
We want to study Zk,n for every possible choice of ti ’s.
Configuration space: PN(R) = {sets of distinct t1, . . . , tN ∈ R}.
Zk,n
PN(R)
Zk,n
(t1, . . . , tN)
Fiber is Zk,n = X (F (t1)) ∩ · · · ∩ X (F (tN)).
Observation: PN(R) is contractible. No monodromy!
If we can label one fiber by tableaux, we can label all of them.
![Page 36: Enumerative Geometry and the Shapiro Shapiro Conjecturesites.math.washington.edu/~jlev/sfu_slides.pdf · Enumerative geometry Some questions about 3D geometry: 1.How many lines meet](https://reader035.vdocument.in/reader035/viewer/2022080717/5f7815655446ce6dd9227922/html5/thumbnails/36.jpg)
Families of geometry problems
We want to study Zk,n for every possible choice of ti ’s.
Configuration space: PN(R) = {sets of distinct t1, . . . , tN ∈ R}.
Zk,n
PN(R)
Zk,n
(t1, . . . , tN)
Fiber is Zk,n = X (F (t1)) ∩ · · · ∩ X (F (tN)).
Observation: PN(R) is contractible. No monodromy!
If we can label one fiber by tableaux, we can label all of them.
![Page 37: Enumerative Geometry and the Shapiro Shapiro Conjecturesites.math.washington.edu/~jlev/sfu_slides.pdf · Enumerative geometry Some questions about 3D geometry: 1.How many lines meet](https://reader035.vdocument.in/reader035/viewer/2022080717/5f7815655446ce6dd9227922/html5/thumbnails/37.jpg)
Families of geometry problems
We want to study Zk,n for every possible choice of ti ’s.
Configuration space: PN(R) = {sets of distinct t1, . . . , tN ∈ R}.
Zk,n
PN(R)
Zk,n
(t1, . . . , tN)
Fiber is Zk,n = X (F (t1)) ∩ · · · ∩ X (F (tN)).
Observation: PN(R) is contractible. No monodromy!
If we can label one fiber by tableaux, we can label all of them.
![Page 38: Enumerative Geometry and the Shapiro Shapiro Conjecturesites.math.washington.edu/~jlev/sfu_slides.pdf · Enumerative geometry Some questions about 3D geometry: 1.How many lines meet](https://reader035.vdocument.in/reader035/viewer/2022080717/5f7815655446ce6dd9227922/html5/thumbnails/38.jpg)
Families of geometry problems
We want to study Zk,n for every possible choice of ti ’s.
Configuration space: PN(R) = {sets of distinct t1, . . . , tN ∈ R}.
Zk,n
PN(R)
Zk,n
(t1, . . . , tN)
Fiber is Zk,n = X (F (t1)) ∩ · · · ∩ X (F (tN)).
Observation: PN(R) is contractible. No monodromy!
If we can label one fiber by tableaux, we can label all of them.
![Page 39: Enumerative Geometry and the Shapiro Shapiro Conjecturesites.math.washington.edu/~jlev/sfu_slides.pdf · Enumerative geometry Some questions about 3D geometry: 1.How many lines meet](https://reader035.vdocument.in/reader035/viewer/2022080717/5f7815655446ce6dd9227922/html5/thumbnails/39.jpg)
How to find combinatorics in geometry
Key idea
Degenerate the problem until it breaks into pieces.
Take (t1, . . . , tN) = (z , z2, . . . , zN) and take limz→0
.
0 1R
z → 0
z1z2z3z4
What will happen to Zk,n at z = 0?
![Page 40: Enumerative Geometry and the Shapiro Shapiro Conjecturesites.math.washington.edu/~jlev/sfu_slides.pdf · Enumerative geometry Some questions about 3D geometry: 1.How many lines meet](https://reader035.vdocument.in/reader035/viewer/2022080717/5f7815655446ce6dd9227922/html5/thumbnails/40.jpg)
How to find combinatorics in geometry
Key idea
Degenerate the problem until it breaks into pieces.
Take (t1, . . . , tN) = (z , z2, . . . , zN) and take limz→0
.
0 1R
z → 0
z1z2z3z4
What will happen to Zk,n at z = 0?
![Page 41: Enumerative Geometry and the Shapiro Shapiro Conjecturesites.math.washington.edu/~jlev/sfu_slides.pdf · Enumerative geometry Some questions about 3D geometry: 1.How many lines meet](https://reader035.vdocument.in/reader035/viewer/2022080717/5f7815655446ce6dd9227922/html5/thumbnails/41.jpg)
How to find combinatorics in geometry
Key idea
Degenerate the problem until it breaks into pieces.
Take (t1, . . . , tN) = (z , z2, . . . , zN) and take limz→0
.
0 1R
z → 0
z1z2z3z4
What will happen to Zk,n at z = 0?
![Page 42: Enumerative Geometry and the Shapiro Shapiro Conjecturesites.math.washington.edu/~jlev/sfu_slides.pdf · Enumerative geometry Some questions about 3D geometry: 1.How many lines meet](https://reader035.vdocument.in/reader035/viewer/2022080717/5f7815655446ce6dd9227922/html5/thumbnails/42.jpg)
Lines through 4 given lines, redux
Note:{
lines in P3}
= Gr(2, 4) ≈{[
0 1 ∗ ∗1 0 ∗ ∗
]}.
Set up Z2,4 using tangent lines from the flags F (t):
Z2,4 = X (F (z)) ∩ · · · ∩ X (F (z4)) ⊂ Gr(2, 4)
={
lines in P3 meeting 4 given (tangent) lines}.
0R
z → 0
Pr (R)
z1z2z3z4
Z2,4 ⊂ Gr(2, 4)
(two solutions)
[0 1 ≈ z ≈ z3
1 0 ≈ z4 ≈ z6
][
0 1 ≈ z ≈ z4
1 0 ≈ z3 ≈ z6
]1 23 4
1 32 4
![Page 43: Enumerative Geometry and the Shapiro Shapiro Conjecturesites.math.washington.edu/~jlev/sfu_slides.pdf · Enumerative geometry Some questions about 3D geometry: 1.How many lines meet](https://reader035.vdocument.in/reader035/viewer/2022080717/5f7815655446ce6dd9227922/html5/thumbnails/43.jpg)
Lines through 4 given lines, redux
Note:{
lines in P3}
= Gr(2, 4) ≈{[
0 1 ∗ ∗1 0 ∗ ∗
]}.
Set up Z2,4 using tangent lines from the flags F (t):
Z2,4 = X (F (z)) ∩ · · · ∩ X (F (z4)) ⊂ Gr(2, 4)
={
lines in P3 meeting 4 given (tangent) lines}.
0R
z → 0
Pr (R)
z1z2z3z4
Z2,4 ⊂ Gr(2, 4)
(two solutions)
[0 1 ≈ z ≈ z3
1 0 ≈ z4 ≈ z6
][
0 1 ≈ z ≈ z4
1 0 ≈ z3 ≈ z6
]1 23 4
1 32 4
![Page 44: Enumerative Geometry and the Shapiro Shapiro Conjecturesites.math.washington.edu/~jlev/sfu_slides.pdf · Enumerative geometry Some questions about 3D geometry: 1.How many lines meet](https://reader035.vdocument.in/reader035/viewer/2022080717/5f7815655446ce6dd9227922/html5/thumbnails/44.jpg)
Lines through 4 given lines, redux
Note:{
lines in P3}
= Gr(2, 4) ≈{[
0 1 ∗ ∗1 0 ∗ ∗
]}.
Set up Z2,4 using tangent lines from the flags F (t):
Z2,4 = X (F (z)) ∩ · · · ∩ X (F (z4)) ⊂ Gr(2, 4)
={
lines in P3 meeting 4 given (tangent) lines}.
0R
z → 0
Pr (R)
z1z2z3z4
Z2,4 ⊂ Gr(2, 4)
(two solutions)
[0 1 ≈ z ≈ z3
1 0 ≈ z4 ≈ z6
][
0 1 ≈ z ≈ z4
1 0 ≈ z3 ≈ z6
]1 23 4
1 32 4
![Page 45: Enumerative Geometry and the Shapiro Shapiro Conjecturesites.math.washington.edu/~jlev/sfu_slides.pdf · Enumerative geometry Some questions about 3D geometry: 1.How many lines meet](https://reader035.vdocument.in/reader035/viewer/2022080717/5f7815655446ce6dd9227922/html5/thumbnails/45.jpg)
Lines through 4 given lines, redux
Note:{
lines in P3}
= Gr(2, 4) ≈{[
0 1 ∗ ∗1 0 ∗ ∗
]}.
Set up Z2,4 using tangent lines from the flags F (t):
Z2,4 = X (F (z)) ∩ · · · ∩ X (F (z4)) ⊂ Gr(2, 4)
={
lines in P3 meeting 4 given (tangent) lines}.
0R
z → 0
Pr (R)
z1z2z3z4
Z2,4 ⊂ Gr(2, 4)
(two solutions)
[0 1 ≈ z ≈ z3
1 0 ≈ z4 ≈ z6
][
0 1 ≈ z ≈ z4
1 0 ≈ z3 ≈ z6
]1 23 4
1 32 4
![Page 46: Enumerative Geometry and the Shapiro Shapiro Conjecturesites.math.washington.edu/~jlev/sfu_slides.pdf · Enumerative geometry Some questions about 3D geometry: 1.How many lines meet](https://reader035.vdocument.in/reader035/viewer/2022080717/5f7815655446ce6dd9227922/html5/thumbnails/46.jpg)
Tableau labels from Plucker coordinates
Limiting matrix:[0 1 ≈ z1 ≈ z3
1 0 ≈ z4 ≈ z6
]1 23 4
Plucker coordinates (minors) on Gr(2, 4):
∅det12 = 1
det13 = O(z1)
det14 = O(z3)
det23 = O(z4)
det24 = O(z6)
det34 = O(z10)
![Page 47: Enumerative Geometry and the Shapiro Shapiro Conjecturesites.math.washington.edu/~jlev/sfu_slides.pdf · Enumerative geometry Some questions about 3D geometry: 1.How many lines meet](https://reader035.vdocument.in/reader035/viewer/2022080717/5f7815655446ce6dd9227922/html5/thumbnails/47.jpg)
Tableau labels from Plucker coordinates
Limiting matrix:[0 1 ≈ z1 ≈ z3
1 0 ≈ z4 ≈ z6
]1 23 4
Plucker coordinates (minors) on Gr(2, 4):
∅ 1
1 2
13
1 23
1 23 4
det12 = z0
det13 = O(z1)
det14 = O(z1+2)
det23 = O(z1+3)
det24 = O(z1+2+3)
det34 = O(z1+2+3+4)
![Page 48: Enumerative Geometry and the Shapiro Shapiro Conjecturesites.math.washington.edu/~jlev/sfu_slides.pdf · Enumerative geometry Some questions about 3D geometry: 1.How many lines meet](https://reader035.vdocument.in/reader035/viewer/2022080717/5f7815655446ce6dd9227922/html5/thumbnails/48.jpg)
Combinatorics and geometry
Theorem. This procedure gives a bijection Zk,n ↔ SYT( ).(Purbhoo ’09, Speyer ’14)
And: moving {ti} on RP1 changes the labels by known algorithms!
RP10 ∞
“tableau promotion”
RP10 ∞
“tableau evacuation”
And more:
I Topology and genus when dim(Z ) = 1 (Levinson, Gillespie–L)
I Orthogonal Grassmannians (Purbhoo, Gillespie–L–Purbhoo)
I Vector bundles on M0,n (Kamnitzer, Rybnikov)
![Page 49: Enumerative Geometry and the Shapiro Shapiro Conjecturesites.math.washington.edu/~jlev/sfu_slides.pdf · Enumerative geometry Some questions about 3D geometry: 1.How many lines meet](https://reader035.vdocument.in/reader035/viewer/2022080717/5f7815655446ce6dd9227922/html5/thumbnails/49.jpg)
Combinatorics and geometry
Theorem. This procedure gives a bijection Zk,n ↔ SYT( ).(Purbhoo ’09, Speyer ’14)
And: moving {ti} on RP1 changes the labels by known algorithms!
RP10 ∞
“tableau promotion”
RP10 ∞
“tableau evacuation”
And more:
I Topology and genus when dim(Z ) = 1 (Levinson, Gillespie–L)
I Orthogonal Grassmannians (Purbhoo, Gillespie–L–Purbhoo)
I Vector bundles on M0,n (Kamnitzer, Rybnikov)
![Page 50: Enumerative Geometry and the Shapiro Shapiro Conjecturesites.math.washington.edu/~jlev/sfu_slides.pdf · Enumerative geometry Some questions about 3D geometry: 1.How many lines meet](https://reader035.vdocument.in/reader035/viewer/2022080717/5f7815655446ce6dd9227922/html5/thumbnails/50.jpg)
Combinatorics and geometry
Theorem. This procedure gives a bijection Zk,n ↔ SYT( ).(Purbhoo ’09, Speyer ’14)
And: moving {ti} on RP1 changes the labels by known algorithms!
RP10 ∞
“tableau promotion”
RP10 ∞
“tableau evacuation”
And more:
I Topology and genus when dim(Z ) = 1 (Levinson, Gillespie–L)
I Orthogonal Grassmannians (Purbhoo, Gillespie–L–Purbhoo)
I Vector bundles on M0,n (Kamnitzer, Rybnikov)
![Page 51: Enumerative Geometry and the Shapiro Shapiro Conjecturesites.math.washington.edu/~jlev/sfu_slides.pdf · Enumerative geometry Some questions about 3D geometry: 1.How many lines meet](https://reader035.vdocument.in/reader035/viewer/2022080717/5f7815655446ce6dd9227922/html5/thumbnails/51.jpg)
Part 3. Topology!
![Page 52: Enumerative Geometry and the Shapiro Shapiro Conjecturesites.math.washington.edu/~jlev/sfu_slides.pdf · Enumerative geometry Some questions about 3D geometry: 1.How many lines meet](https://reader035.vdocument.in/reader035/viewer/2022080717/5f7815655446ce6dd9227922/html5/thumbnails/52.jpg)
A challenge and a new approach
Theorem (M–T–V ’05, ’09)
For t1, . . . , tN ∈ RP1, Zk,n consists of real, multiplicity-free points.
Challenge for geometers:
I M–T–V proof uses integrable systems, the Bethe ansatz
I Subsequent geometry work used M–T–V as black box.
I Many open generalizations of interest!
It turns out there is a topological / geometric approach.
(+)1 2 5
3 4 6, (−)
1 3 5
2 4 6, · · ·
Oriented Young tableaux.
![Page 53: Enumerative Geometry and the Shapiro Shapiro Conjecturesites.math.washington.edu/~jlev/sfu_slides.pdf · Enumerative geometry Some questions about 3D geometry: 1.How many lines meet](https://reader035.vdocument.in/reader035/viewer/2022080717/5f7815655446ce6dd9227922/html5/thumbnails/53.jpg)
A challenge and a new approach
Theorem (M–T–V ’05, ’09)
For t1, . . . , tN ∈ RP1, Zk,n consists of real, multiplicity-free points.
Challenge for geometers:
I M–T–V proof uses integrable systems, the Bethe ansatz
I Subsequent geometry work used M–T–V as black box.
I Many open generalizations of interest!
It turns out there is a topological / geometric approach.
(+)1 2 5
3 4 6, (−)
1 3 5
2 4 6, · · ·
Oriented Young tableaux.
![Page 54: Enumerative Geometry and the Shapiro Shapiro Conjecturesites.math.washington.edu/~jlev/sfu_slides.pdf · Enumerative geometry Some questions about 3D geometry: 1.How many lines meet](https://reader035.vdocument.in/reader035/viewer/2022080717/5f7815655446ce6dd9227922/html5/thumbnails/54.jpg)
A challenge and a new approach
Theorem (M–T–V ’05, ’09)
For t1, . . . , tN ∈ RP1, Zk,n consists of real, multiplicity-free points.
Challenge for geometers:
I M–T–V proof uses integrable systems, the Bethe ansatz
I Subsequent geometry work used M–T–V as black box.
I Many open generalizations of interest!
It turns out there is a topological / geometric approach.
(+)1 2 5
3 4 6, (−)
1 3 5
2 4 6, · · ·
Oriented Young tableaux.
![Page 55: Enumerative Geometry and the Shapiro Shapiro Conjecturesites.math.washington.edu/~jlev/sfu_slides.pdf · Enumerative geometry Some questions about 3D geometry: 1.How many lines meet](https://reader035.vdocument.in/reader035/viewer/2022080717/5f7815655446ce6dd9227922/html5/thumbnails/55.jpg)
Generalization: complex conjugate points on P1
Before: Defined Zk,n using real points ti ∈ RP1.
Now: Zk,n =
n1⋂i=1
X (ti )︸ ︷︷ ︸real
∩n2⋂j=1
X (tj) ∩ X (tj)︸ ︷︷ ︸complex conjugate pairs
.
Mixed configuration space:For a partition µ = (2n2 , 1n1), let
P(µ) =
{sets of
n1 distinct points on R,n2 complex conjugate pairs on C \ R
}⊆ Pr (C) (r = 2n2 + n1).
(Base case: µ = (1N), all real ti .)
![Page 56: Enumerative Geometry and the Shapiro Shapiro Conjecturesites.math.washington.edu/~jlev/sfu_slides.pdf · Enumerative geometry Some questions about 3D geometry: 1.How many lines meet](https://reader035.vdocument.in/reader035/viewer/2022080717/5f7815655446ce6dd9227922/html5/thumbnails/56.jpg)
Generalization: complex conjugate points on P1
Before: Defined Zk,n using real points ti ∈ RP1.
Now: Zk,n =
n1⋂i=1
X (ti )︸ ︷︷ ︸real
∩n2⋂j=1
X (tj) ∩ X (tj)︸ ︷︷ ︸complex conjugate pairs
.
Mixed configuration space:For a partition µ = (2n2 , 1n1), let
P(µ) =
{sets of
n1 distinct points on R,n2 complex conjugate pairs on C \ R
}⊆ Pr (C) (r = 2n2 + n1).
(Base case: µ = (1N), all real ti .)
![Page 57: Enumerative Geometry and the Shapiro Shapiro Conjecturesites.math.washington.edu/~jlev/sfu_slides.pdf · Enumerative geometry Some questions about 3D geometry: 1.How many lines meet](https://reader035.vdocument.in/reader035/viewer/2022080717/5f7815655446ce6dd9227922/html5/thumbnails/57.jpg)
Generalization: complex conjugate points on P1
Before: Defined Zk,n using real points ti ∈ RP1.
Now: Zk,n =
n1⋂i=1
X (ti )︸ ︷︷ ︸real
∩n2⋂j=1
X (tj) ∩ X (tj)︸ ︷︷ ︸complex conjugate pairs
.
Mixed configuration space:For a partition µ = (2n2 , 1n1), let
P(µ) =
{sets of
n1 distinct points on R,n2 complex conjugate pairs on C \ R
}⊆ Pr (C) (r = 2n2 + n1).
(Base case: µ = (1N), all real ti .)
![Page 58: Enumerative Geometry and the Shapiro Shapiro Conjecturesites.math.washington.edu/~jlev/sfu_slides.pdf · Enumerative geometry Some questions about 3D geometry: 1.How many lines meet](https://reader035.vdocument.in/reader035/viewer/2022080717/5f7815655446ce6dd9227922/html5/thumbnails/58.jpg)
Topological and algebraic degrees
How many real points in Zk,n for (t1, . . . , tN) ∈ P(µ)?
I Upper bound from (algebraic) degree = #SYT( ).
I Lower bound from topological degree... (=?):
Zk,n
P(µ)
+
−
+
+
+
Algebraic degree: 3
Topological degree: 1
I We use a careful twist of standard orientation on Gr(k, n).
![Page 59: Enumerative Geometry and the Shapiro Shapiro Conjecturesites.math.washington.edu/~jlev/sfu_slides.pdf · Enumerative geometry Some questions about 3D geometry: 1.How many lines meet](https://reader035.vdocument.in/reader035/viewer/2022080717/5f7815655446ce6dd9227922/html5/thumbnails/59.jpg)
Topological and algebraic degrees
How many real points in Zk,n for (t1, . . . , tN) ∈ P(µ)?
I Upper bound from (algebraic) degree = #SYT( ).
I Lower bound from topological degree... (=?):
Zk,n
P(µ)
+
−
+
+
+
Algebraic degree: 3
Topological degree: 1
I We use a careful twist of standard orientation on Gr(k, n).
![Page 60: Enumerative Geometry and the Shapiro Shapiro Conjecturesites.math.washington.edu/~jlev/sfu_slides.pdf · Enumerative geometry Some questions about 3D geometry: 1.How many lines meet](https://reader035.vdocument.in/reader035/viewer/2022080717/5f7815655446ce6dd9227922/html5/thumbnails/60.jpg)
Topological and algebraic degrees
How many real points in Zk,n for (t1, . . . , tN) ∈ P(µ)?
I Upper bound from (algebraic) degree = #SYT( ).
I Lower bound from topological degree... (=?):
Zk,n
P(µ)
+
−
+
+
+
Algebraic degree: 3
Topological degree: 1
I We use a careful twist of standard orientation on Gr(k, n).
![Page 61: Enumerative Geometry and the Shapiro Shapiro Conjecturesites.math.washington.edu/~jlev/sfu_slides.pdf · Enumerative geometry Some questions about 3D geometry: 1.How many lines meet](https://reader035.vdocument.in/reader035/viewer/2022080717/5f7815655446ce6dd9227922/html5/thumbnails/61.jpg)
Topological and algebraic degrees
How many real points in Zk,n for (t1, . . . , tN) ∈ P(µ)?
I Upper bound from (algebraic) degree = #SYT( ).
I Lower bound from topological degree... (=?):
Zk,n
P(µ)
+
−
+
+
+
Algebraic degree: 3Topological degree: 1
I We use a careful twist of standard orientation on Gr(k, n).
![Page 62: Enumerative Geometry and the Shapiro Shapiro Conjecturesites.math.washington.edu/~jlev/sfu_slides.pdf · Enumerative geometry Some questions about 3D geometry: 1.How many lines meet](https://reader035.vdocument.in/reader035/viewer/2022080717/5f7815655446ce6dd9227922/html5/thumbnails/62.jpg)
Topological and algebraic degrees
How many real points in Zk,n for (t1, . . . , tN) ∈ P(µ)?
I Upper bound from (algebraic) degree = #SYT( ).
I Lower bound from topological degree... (=?):
Zk,n
P(µ)
+
−
+
+
+
Algebraic degree: 3Topological degree: 1
I We use a careful twist of standard orientation on Gr(k, n).
![Page 63: Enumerative Geometry and the Shapiro Shapiro Conjecturesites.math.washington.edu/~jlev/sfu_slides.pdf · Enumerative geometry Some questions about 3D geometry: 1.How many lines meet](https://reader035.vdocument.in/reader035/viewer/2022080717/5f7815655446ce6dd9227922/html5/thumbnails/63.jpg)
The topological degree of Zk ,n
Character table of S4. (χλ(µ))
λ, µ (4) (3, 1) (22) (2, 12) (14)
1 1 1 1 1
1 0 -1 -1 3
0 -1 2 0 2
-1 0 -1 1 3
-1 1 1 -1 1
For Sk(n−k), let be the k × (n − k) rectangle, µ = (2n2 , 1n1).
Theorem (L, Purbhoo ‘19)
There is an orientation, the character orientation, such that thefamily Zk,n has topological degree χ (µ) over P(µ).
![Page 64: Enumerative Geometry and the Shapiro Shapiro Conjecturesites.math.washington.edu/~jlev/sfu_slides.pdf · Enumerative geometry Some questions about 3D geometry: 1.How many lines meet](https://reader035.vdocument.in/reader035/viewer/2022080717/5f7815655446ce6dd9227922/html5/thumbnails/64.jpg)
The topological degree of Zk ,n
Character table of S4. (χλ(µ))
λ, µ (4) (3, 1) (22) (2, 12) (14)
1 1 1 1 1
1 0 -1 -1 3
0 -1 2 0 2
-1 0 -1 1 3
-1 1 1 -1 1
For Sk(n−k), let be the k × (n − k) rectangle, µ = (2n2 , 1n1).
Theorem (L, Purbhoo ‘19)
There is an orientation, the character orientation, such that thefamily Zk,n has topological degree χ (µ) over P(µ).
![Page 65: Enumerative Geometry and the Shapiro Shapiro Conjecturesites.math.washington.edu/~jlev/sfu_slides.pdf · Enumerative geometry Some questions about 3D geometry: 1.How many lines meet](https://reader035.vdocument.in/reader035/viewer/2022080717/5f7815655446ce6dd9227922/html5/thumbnails/65.jpg)
The topological degree of Zk ,n
Character table of S4. (χλ(µ))
λ, µ (4) (3, 1) (22) (2, 12) (14)
1 1 1 1 1
1 0 -1 -1 3
0 -1 2 0 2
-1 0 -1 1 3
-1 1 1 -1 1
For Sk(n−k), let be the k × (n − k) rectangle, µ = (2n2 , 1n1).
Theorem (L, Purbhoo ‘19)
There is an orientation, the character orientation, such that thefamily Zk,n has topological degree χ (µ) over P(µ).
![Page 66: Enumerative Geometry and the Shapiro Shapiro Conjecturesites.math.washington.edu/~jlev/sfu_slides.pdf · Enumerative geometry Some questions about 3D geometry: 1.How many lines meet](https://reader035.vdocument.in/reader035/viewer/2022080717/5f7815655446ce6dd9227922/html5/thumbnails/66.jpg)
Signed Young tableaux
Theorem (L, Purbhoo ‘19)
There is an orientation, the character orientation, such that thefamily Zk,n has topological degree χ (µ) over P(µ).
Murnaghan–Nakayama rule for χλ(µ), µ = (2n2 , 1n1):
χλ(µ) =∑T
(−1)# (T ) :µ-domino tableaux (+)
1 2 4
3 5 6, (−)
1 3 4
2 5 6, · · ·
shape(T ) = λ.
I Special case: µ = (1N), no dominos χ (1N) = #SYT.
I Corollary: Shapiro–Shapiro Conjecture.
![Page 67: Enumerative Geometry and the Shapiro Shapiro Conjecturesites.math.washington.edu/~jlev/sfu_slides.pdf · Enumerative geometry Some questions about 3D geometry: 1.How many lines meet](https://reader035.vdocument.in/reader035/viewer/2022080717/5f7815655446ce6dd9227922/html5/thumbnails/67.jpg)
Signed Young tableaux
Theorem (L, Purbhoo ‘19)
There is an orientation, the character orientation, such that thefamily Zk,n has topological degree χ (µ) over P(µ).
Murnaghan–Nakayama rule for χλ(µ), µ = (2n2 , 1n1):
χλ(µ) =∑T
(−1)# (T ) :µ-domino tableaux (+)
1 2 4
3 5 6, (−)
1 3 4
2 5 6, · · ·
shape(T ) = λ.
I Special case: µ = (1N), no dominos χ (1N) = #SYT.
I Corollary: Shapiro–Shapiro Conjecture.
![Page 68: Enumerative Geometry and the Shapiro Shapiro Conjecturesites.math.washington.edu/~jlev/sfu_slides.pdf · Enumerative geometry Some questions about 3D geometry: 1.How many lines meet](https://reader035.vdocument.in/reader035/viewer/2022080717/5f7815655446ce6dd9227922/html5/thumbnails/68.jpg)
Signed Young tableaux
Theorem (L, Purbhoo ‘19)
There is an orientation, the character orientation, such that thefamily Zk,n has topological degree χ (µ) over P(µ).
Murnaghan–Nakayama rule for χλ(µ), µ = (2n2 , 1n1):
χλ(µ) =∑T
(−1)# (T ) :µ-domino tableaux (+)
1 2 4
3 5 6, (−)
1 3 4
2 5 6, · · ·
shape(T ) = λ.
I Special case: µ = (1N), no dominos χ (1N) = #SYT.
I Corollary: Shapiro–Shapiro Conjecture.
![Page 69: Enumerative Geometry and the Shapiro Shapiro Conjecturesites.math.washington.edu/~jlev/sfu_slides.pdf · Enumerative geometry Some questions about 3D geometry: 1.How many lines meet](https://reader035.vdocument.in/reader035/viewer/2022080717/5f7815655446ce6dd9227922/html5/thumbnails/69.jpg)
Labeling Zk ,n by signed Young tableaux
Proof sketch:
I Label limit fibers by tableaux.
I Track +/− signs along a network of paths.
Case 1: 1
2←→ 1
2/ 1 2 ←→ 1 2
++ + + + +
++
+−
− −
P(µ = (16)) P(µ′ = (2, 14))
P(µ) P(µ′)
Z
Z1 2 43 5 6
1 3 42 5 6
1 2 4
3 5 6
1 3 4
2 5 6
(+1)
(−1)
![Page 70: Enumerative Geometry and the Shapiro Shapiro Conjecturesites.math.washington.edu/~jlev/sfu_slides.pdf · Enumerative geometry Some questions about 3D geometry: 1.How many lines meet](https://reader035.vdocument.in/reader035/viewer/2022080717/5f7815655446ce6dd9227922/html5/thumbnails/70.jpg)
Labeling Zk ,n by signed Young tableaux
Proof sketch:
I Label limit fibers by tableaux.
I Track +/− signs along a network of paths.
Case 1: 1
2←→ 1
2/ 1 2 ←→ 1 2
++ + + + +
++
+−
− −
P(µ = (16)) P(µ′ = (2, 14))
P(µ) P(µ′)
Z
Z1 2 43 5 6
1 3 42 5 6
1 2 4
3 5 6
1 3 4
2 5 6
(+1)
(−1)
![Page 71: Enumerative Geometry and the Shapiro Shapiro Conjecturesites.math.washington.edu/~jlev/sfu_slides.pdf · Enumerative geometry Some questions about 3D geometry: 1.How many lines meet](https://reader035.vdocument.in/reader035/viewer/2022080717/5f7815655446ce6dd9227922/html5/thumbnails/71.jpg)
Labeling Zk ,n by signed Young tableaux
Proof sketch:
I Label limit fibers by tableaux.
I Track +/− signs along a network of paths.
Case 2: 32 ←→ 2
3
+ + + + +
++
+−
−
P(µ = (16)) P(µ = (2, 14))
P(µ) P(µ′)
Z
Z
1 3 42 5 6
1 2 43 5 6
(∅) (+1)
![Page 72: Enumerative Geometry and the Shapiro Shapiro Conjecturesites.math.washington.edu/~jlev/sfu_slides.pdf · Enumerative geometry Some questions about 3D geometry: 1.How many lines meet](https://reader035.vdocument.in/reader035/viewer/2022080717/5f7815655446ce6dd9227922/html5/thumbnails/72.jpg)
Open questions
I (Representation theory).Do all SN characters χλ(µ) give topological degrees of realSchubert problems?
I (Complex geometry).Explicit geometry over P(µ) for µ 6= (1N)?
I (Stable curves).How does the geometry look over the moduli space M0,N?
I M0,N(R) is non-orientable!
Many interesting relationships to find between geometry andcombinatorics.
![Page 73: Enumerative Geometry and the Shapiro Shapiro Conjecturesites.math.washington.edu/~jlev/sfu_slides.pdf · Enumerative geometry Some questions about 3D geometry: 1.How many lines meet](https://reader035.vdocument.in/reader035/viewer/2022080717/5f7815655446ce6dd9227922/html5/thumbnails/73.jpg)
Thank you!