Association with Interpolation Aaron Landesman (Stanford University) Anand Patel (Harvard University) Joint Mathematics Meetings Atlanta, GA January 6, 2017
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
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
Interpolation of Nonspecial curves Theorem (Atanasov) Nonspecial curves of degree d, genus g in P 3 satisfy interpolation if d ≥ g + 3 unless ( d , g , r ) = (5 , 2 , 3) . Aaron Landesman and Anand Patel Association with Interpolation 4 / 21
Interpolation of Nonspecial curves Theorem (Atanasov) Nonspecial curves of degree d, genus g in P 3 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 P r 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
Friedman’s Conference Perspectives on Complex Algebraic Geometry SPEAKERS Celebrating the achievements and influence of Robert Friedman D. Arapura (Purdue) on the occasion of his 60th birthday A. Beauville (Universite de Nice) F. Catanese (Universitat Bayreuth) Columbia University Department of Mathematics H. Clemens (Ohio State) May 22-25, 2015 R. Donagi (U Penn) https://sites.google.com/site/complexalgebraicgeometry/ S. Donaldson (Imperial College) Funded by the NSF and Columbia University 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
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
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 P 7 ? Aaron Landesman and Anand Patel Association with Interpolation 6 / 21
Varieties of Minimal Degree Definition A variety of dimension k and degree d in P r 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
Results Theorem (L–) Varieties of minimal degree satisfy interpolation. Aaron Landesman and Anand Patel Association with Interpolation 8 / 21
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
Rational Normal Curves Example In dimension k = 1, interpolation of varieties of minimal degree says that through n + 3 points in P n there passes a rational normal curve. Aaron Landesman and Anand Patel Association with Interpolation 9 / 21
Rational Normal Curves Example In dimension k = 1, interpolation of varieties of minimal degree says that through n + 3 points in P n 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 P 2 there passes a plane conic. Aaron Landesman and Anand Patel Association with Interpolation 9 / 21
Conic Interpolation Aaron Landesman and Anand Patel Association with Interpolation 10 / 21
Conic Interpolation Aaron Landesman and Anand Patel Association with Interpolation 11 / 21
Conic Interpolation Aaron Landesman and Anand Patel Association with Interpolation 12 / 21
Conic Interpolation Aaron Landesman and Anand Patel Association with Interpolation 13 / 21
Conic Interpolation Aaron Landesman and Anand Patel Association with Interpolation 14 / 21
3-Veronese Interpolation Definition A 3 -Veronese surface is, up to linear change of variables, a copy of P 2 embedded by degree 3 polynomials P 2 → P 9 x 3 : x 2 y : x 2 z : xy 2 : xyz : xz 2 : y 3 : y 2 z : yz 2 : z 3 � � [ x : y : z ] �→ . Example Interpolation says that through 13 points in P 9 there passes a 3-Veronese surface. Aaron Landesman and Anand Patel Association with Interpolation 15 / 21
3-Veronese Interpolation Aaron Landesman and Anand Patel Association with Interpolation 16 / 21
3-Veronese Interpolation Aaron Landesman and Anand Patel Association with Interpolation 17 / 21
3-Veronese Interpolation Aaron Landesman and Anand Patel Association with Interpolation 18 / 21
3-Veronese Interpolation Aaron Landesman and Anand Patel Association with Interpolation 19 / 21
3-Veronese Interpolation Aaron Landesman and Anand Patel Association with Interpolation 20 / 21
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
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 P 6 , 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
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