Association with Interpolation Aaron Landesman (Stanford University) - - PowerPoint PPT Presentation

▶

Mar 25, 2024 337 likes •609 views

Association with Interpolation Aaron Landesman (Stanford University) Anand Patel (Harvard University) Joint Mathematics Meetings Atlanta, GA January 6, 2017 Euclids Postulates Figure : Euclids Postulate 5: through any two points there

SLIDE 1

Association with Interpolation

Aaron Landesman (Stanford University) Anand Patel (Harvard University) Joint Mathematics Meetings Atlanta, GA January 6, 2017

SLIDE 2

Euclid’s Postulates

Figure : Euclid’s Postulate 5: through any two points there passes a line

Aaron Landesman and Anand Patel Association with Interpolation 2 / 21

SLIDE 3

Definition of Interpolation

An interpolation problem involves two pieces of data:

1 A class H of varieties in projective space (e.g. “lines in the plane”). 2 A collection of incidence conditions (e.g. “passing through two fixed

points”).

Question (Interpolation)

Is there a variety [X] ∈ H meeting a general choice of conditions of the specified type?

Definition

We say H satisfies interpolation if there is [X] ∈ H passing through the maximum possible number of general points.

Aaron Landesman and Anand Patel Association with Interpolation 3 / 21

SLIDE 4

Interpolation of Nonspecial curves

Theorem (Atanasov)

Nonspecial curves of degree d, genus g in P3 satisfy interpolation if d ≥ g + 3 unless (d, g, r) = (5, 2, 3).

Aaron Landesman and Anand Patel Association with Interpolation 4 / 21

SLIDE 5

Interpolation of Nonspecial curves

Theorem (Atanasov)

Nonspecial curves of degree d, genus g in P3 satisfy interpolation if d ≥ g + 3 unless (d, g, r) = (5, 2, 3).

Theorem (Atanasov-Larson-Yang)

Nonspecial curves of degree d, genus g in Pr satisfy interpolation if d ≥ g + r unless (d, g, r) ∈ {(5, 2, 3), (6, 2, 4), (7, 2, 5)} .

Aaron Landesman and Anand Patel Association with Interpolation 4 / 21

SLIDE 6

Friedman’s Conference

Perspectives on Complex Algebraic Geometry

Celebrating the achievements and influence of Robert Friedman

n the occasion of his 60th birthday

Columbia University

Department of Mathematics May 22-25, 2015

https://sites.google.com/site/complexalgebraicgeometry/ Funded by the NSF and Columbia University

SPEAKERS

D. Arapura (Purdue)
A. Beauville (Universite de Nice)
F. Catanese (Universitat Bayreuth)
H. Clemens (Ohio State)
R. Donagi (U Penn)
S. Donaldson (Imperial College)
P. Engel (Columbia)
P. Griffiths (IAS)
J. Harris (Harvard)
D. Huybrechts (Universitat Bonn)
R. Laza (Stony Brook)
E. Looijenga (Utrecht and Tsinghua)
R. Miranda (Colorado State)
J. Morgan (Simons Center)
D. Morrison (UC Santa Barbara)
N. Shepherd-Barron (Cambridge)
E. Witten (IAS)

Organizers:

S. Casalaina-Martin
J. de Jong
R. Laza
J. Morgan
M. Thaddeus

Aaron Landesman and Anand Patel Association with Interpolation 5 / 21

SLIDE 7

Canonical Curves

Theorem (Stevens, 1989)

Canonical curves of genus 4 and 6 do not satisfy interpolation, while canonical curves of genus 3, 5, 7, and g ≥ 9 satisfy interpolation.

Aaron Landesman and Anand Patel Association with Interpolation 6 / 21

SLIDE 8

Canonical Curves

Theorem (Stevens, 1989)

Canonical curves of genus 4 and 6 do not satisfy interpolation, while canonical curves of genus 3, 5, 7, and g ≥ 9 satisfy interpolation.

Question

What about genus 8? That is, is there a genus 8 canonical curve passing through 14 points in P7?

Aaron Landesman and Anand Patel Association with Interpolation 6 / 21

SLIDE 9

Varieties of Minimal Degree

Definition

A variety of dimension k and degree d in Pr is of minimal degree if d = r + 1 − k and of almost minimal degree if d = r + 2 − k.

Theorem (Coskun, 2006)

Surfaces of minimal degree satisfy interpolation.

Question

Do higher dimensional varieties of minimal degree satisfy interpolation? Do surfaces of almost minimal degree satisfy interpolation?

Aaron Landesman and Anand Patel Association with Interpolation 7 / 21

SLIDE 10

Results

Theorem (L–)

Varieties of minimal degree satisfy interpolation.

Aaron Landesman and Anand Patel Association with Interpolation 8 / 21

SLIDE 11

Results

Theorem (L–)

Varieties of minimal degree satisfy interpolation.

Theorem (L–, Patel)

Surfaces of almost minimal degree satisfy interpolation.

Aaron Landesman and Anand Patel Association with Interpolation 8 / 21

SLIDE 12

Rational Normal Curves

Example

In dimension k = 1, interpolation of varieties of minimal degree says that through n + 3 points in Pn there passes a rational normal curve.

Aaron Landesman and Anand Patel Association with Interpolation 9 / 21

SLIDE 13

Rational Normal Curves

Example

In dimension k = 1, interpolation of varieties of minimal degree says that through n + 3 points in Pn there passes a rational normal curve.

Example

In dimension k = 1 and degree d = 2, interpolation of varieties of minimal degree says that through 5 points in P2 there passes a plane conic.

Aaron Landesman and Anand Patel Association with Interpolation 9 / 21

SLIDE 14

Conic Interpolation

Aaron Landesman and Anand Patel Association with Interpolation 10 / 21

SLIDE 15

Conic Interpolation

Aaron Landesman and Anand Patel Association with Interpolation 11 / 21

SLIDE 16

Conic Interpolation

Aaron Landesman and Anand Patel Association with Interpolation 12 / 21

SLIDE 17

Conic Interpolation

Aaron Landesman and Anand Patel Association with Interpolation 13 / 21

SLIDE 18

Conic Interpolation

Aaron Landesman and Anand Patel Association with Interpolation 14 / 21

SLIDE 19

3-Veronese Interpolation

Definition

A 3-Veronese surface is, up to linear change of variables, a copy of P2 embedded by degree 3 polynomials P2 → P9 [x : y : z] →

x3 : x2y : x2z : xy2 : xyz : xz2 : y3 : y2z : yz2 : z3

.

Example

Interpolation says that through 13 points in P9 there passes a 3-Veronese surface.

Aaron Landesman and Anand Patel Association with Interpolation 15 / 21

SLIDE 20

3-Veronese Interpolation

Aaron Landesman and Anand Patel Association with Interpolation 16 / 21

SLIDE 21

3-Veronese Interpolation

Aaron Landesman and Anand Patel Association with Interpolation 17 / 21

SLIDE 22

3-Veronese Interpolation

Aaron Landesman and Anand Patel Association with Interpolation 18 / 21

SLIDE 23

3-Veronese Interpolation

Aaron Landesman and Anand Patel Association with Interpolation 19 / 21

SLIDE 24

3-Veronese Interpolation

Aaron Landesman and Anand Patel Association with Interpolation 20 / 21

SLIDE 25

Canonical Curves

Theorem (Stevens, 1989)

Canonical curves of genus 4 and 6 do not satisfy interpolation, while canonical curves of genus 3, 5, 7, and g ≥ 9 satisfy interpolation.

Aaron Landesman and Anand Patel Association with Interpolation 21 / 21

SLIDE 26

Canonical Curves

Theorem (Stevens, 1989)

Canonical curves of genus 4 and 6 do not satisfy interpolation, while canonical curves of genus 3, 5, 7, and g ≥ 9 satisfy interpolation.

Theorem (Stevens, 1996)

Canonical curves of genus 8 satisfy interpolation.

Proof.

Use association to reduce to a question about smoothability of a cone of lines over 14 points in P6, and then use the computer to check that 14 lines in (Z/31991)6 is smoothable!

Aaron Landesman and Anand Patel Association with Interpolation 21 / 21