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Balinskis theorem and Regularity of Line Arrangements Bruno Benedetti (University of Miami) CombinaTeXas, May 7, 2016 Bruno Benedetti (University of Miami) Balinskis theorem and Regularity of Line Arrangements Joint work with Bruno

  1. Plan for today Part I. Many Classes of Dual Graphs. Part II. Some Algebraic Machinery. Bruno Benedetti (University of Miami) Balinski’s theorem and Regularity of Line Arrangements

  2. Plan for today Part I. Many Classes of Dual Graphs. Part II. Some Algebraic Machinery. Part III. (time permitting) Arrangements of Curves. Bruno Benedetti (University of Miami) Balinski’s theorem and Regularity of Line Arrangements

  3. Many Classes of Dual graphs Bruno Benedetti (University of Miami) Balinski’s theorem and Regularity of Line Arrangements

  4. Many Classes of Dual graphs Of (pure) simplicial complexes (e.g. polytope boundaries): Bruno Benedetti (University of Miami) Balinski’s theorem and Regularity of Line Arrangements

  5. Many Classes of Dual graphs Of (pure) simplicial complexes (e.g. polytope boundaries): Of arrangements of lines or of curves : Bruno Benedetti (University of Miami) Balinski’s theorem and Regularity of Line Arrangements

  6. Many Classes of Dual graphs Of (pure) simplicial complexes (e.g. polytope boundaries): Of arrangements of lines or of curves : (There’s also a “dual multigraph” model, keeping track on how many intersections, with multiple edges/loops.) Bruno Benedetti (University of Miami) Balinski’s theorem and Regularity of Line Arrangements

  7. Many Classes of Dual graphs Of (pure) simplicial complexes (e.g. polytope boundaries): Of arrangements of lines or of curves : (There’s also a “dual multigraph” model, keeping track on how many intersections, with multiple edges/loops.) Of (equidimensional) subspace arrangements or algebraic varieties : Bruno Benedetti (University of Miami) Balinski’s theorem and Regularity of Line Arrangements

  8. Many Classes of Dual graphs Of (pure) simplicial complexes (e.g. polytope boundaries): Of arrangements of lines or of curves : (There’s also a “dual multigraph” model, keeping track on how many intersections, with multiple edges/loops.) Of (equidimensional) subspace arrangements or algebraic varieties : Bruno Benedetti (University of Miami) Balinski’s theorem and Regularity of Line Arrangements

  9. Many Classes of Dual graphs Of (pure) simplicial complexes (e.g. polytope boundaries): Of arrangements of lines or of curves : (There’s also a “dual multigraph” model, keeping track on how many intersections, with multiple edges/loops.) Of (equidimensional) subspace arrangements or algebraic varieties : Vertices correspond to the irreducible components C 1 , . . . , C s . Bruno Benedetti (University of Miami) Balinski’s theorem and Regularity of Line Arrangements

  10. Many Classes of Dual graphs Of (pure) simplicial complexes (e.g. polytope boundaries): Of arrangements of lines or of curves : (There’s also a “dual multigraph” model, keeping track on how many intersections, with multiple edges/loops.) Of (equidimensional) subspace arrangements or algebraic varieties : Vertices correspond to the irreducible components C 1 , . . . , C s . (Equidimensional means, they all have same dimension.) Bruno Benedetti (University of Miami) Balinski’s theorem and Regularity of Line Arrangements

  11. Many Classes of Dual graphs Of (pure) simplicial complexes (e.g. polytope boundaries): Of arrangements of lines or of curves : (There’s also a “dual multigraph” model, keeping track on how many intersections, with multiple edges/loops.) Of (equidimensional) subspace arrangements or algebraic varieties : Vertices correspond to the irreducible components C 1 , . . . , C s . (Equidimensional means, they all have same dimension.) We put an edge between two distinct vertices, if and only if the corresponding components intersect in dimension one less. Bruno Benedetti (University of Miami) Balinski’s theorem and Regularity of Line Arrangements

  12. Dual graphs of curves = dual graphs of varieties By intersecting a d -dimensional object in P n with a generic hyperplane, we get an object in P n − 1 with dimension d − 1, and same dual graph! Bruno Benedetti (University of Miami) Balinski’s theorem and Regularity of Line Arrangements

  13. Dual graphs of curves = dual graphs of varieties By intersecting a d -dimensional object in P n with a generic hyperplane, we get an object in P n − 1 with dimension d − 1, and same dual graph! This way you can always reduce yourself to an (algebraic) curve arrangement with same dual graph. Bruno Benedetti (University of Miami) Balinski’s theorem and Regularity of Line Arrangements

  14. Dual graphs of curves = dual graphs of varieties By intersecting a d -dimensional object in P n with a generic hyperplane, we get an object in P n − 1 with dimension d − 1, and same dual graph! This way you can always reduce yourself to an (algebraic) curve arrangement with same dual graph. picture from mathwarehouse.com Bruno Benedetti (University of Miami) Balinski’s theorem and Regularity of Line Arrangements

  15. Dual graphs of curves = dual graphs of varieties By intersecting a d -dimensional object in P n with a generic hyperplane, we get an object in P n − 1 with dimension d − 1, and same dual graph! This way you can always reduce yourself to an (algebraic) curve arrangement with same dual graph. picture from mathwarehouse.com NOTE: If you started with an arrangement of hyperplanes (or of linear subspaces), you end up with an arrangement of lines . Bruno Benedetti (University of Miami) Balinski’s theorem and Regularity of Line Arrangements

  16. Dual graphs of curves = dual graphs of varieties By intersecting a d -dimensional object in P n with a generic hyperplane, we get an object in P n − 1 with dimension d − 1, and same dual graph! This way you can always reduce yourself to an (algebraic) curve arrangement with same dual graph. picture from mathwarehouse.com NOTE: If you started with an arrangement of hyperplanes (or of linear subspaces), you end up with an arrangement of lines .   � dual graphs dual graphs �    = of subspace of lines arr’ts  Bruno Benedetti (University of Miami) Balinski’s theorem and Regularity of Line Arrangements

  17. Dual graphs of curves = dual graphs of varieties By intersecting a d -dimensional object in P n with a generic hyperplane, we get an object in P n − 1 with dimension d − 1, and same dual graph! This way you can always reduce yourself to an (algebraic) curve arrangement with same dual graph. picture from mathwarehouse.com NOTE: If you started with an arrangement of hyperplanes (or of linear subspaces), you end up with an arrangement of lines .   � dual graphs � dual graphs dual graphs � �    = ⊂ of subspace of lines of curves arr’ts  Bruno Benedetti (University of Miami) Balinski’s theorem and Regularity of Line Arrangements

  18. Dual graphs of curves = dual graphs of varieties By intersecting a d -dimensional object in P n with a generic hyperplane, we get an object in P n − 1 with dimension d − 1, and same dual graph! This way you can always reduce yourself to an (algebraic) curve arrangement with same dual graph. picture from mathwarehouse.com NOTE: If you started with an arrangement of hyperplanes (or of linear subspaces), you end up with an arrangement of lines .   � dual graphs � dual graphs � dual graphs dual graphs � � �    = ⊂ = . of subspace of lines of curves of varieties arr’ts  Bruno Benedetti (University of Miami) Balinski’s theorem and Regularity of Line Arrangements

  19. Not all graphs are dual to a line arrangement Attention!, graphs like G 0 = { 12 , 34 } ∪ { 15 , 25 , 35 , 45 } ∪ { 16 , 26 , 36 , 46 } ∪ { 17 , 27 , 37 , 47 } are not dual to any Euclidean line arrangement! Bruno Benedetti (University of Miami) Balinski’s theorem and Regularity of Line Arrangements

  20. Not all graphs are dual to a line arrangement Attention!, graphs like G 0 = { 12 , 34 } ∪ { 15 , 25 , 35 , 45 } ∪ { 16 , 26 , 36 , 46 } ∪ { 17 , 27 , 37 , 47 } are not dual to any Euclidean line arrangement! Try drawing it. Let P = r 1 ∩ r 2 and let Q = r 3 ∩ r 4 . Bruno Benedetti (University of Miami) Balinski’s theorem and Regularity of Line Arrangements

  21. Not all graphs are dual to a line arrangement Attention!, graphs like G 0 = { 12 , 34 } ∪ { 15 , 25 , 35 , 45 } ∪ { 16 , 26 , 36 , 46 } ∪ { 17 , 27 , 37 , 47 } are not dual to any Euclidean line arrangement! Try drawing it. Let P = r 1 ∩ r 2 and let Q = r 3 ∩ r 4 . Let p be the plane containing r 1 ∪ r 2 , and let q be the plane containing r 3 ∪ r 4 . Bruno Benedetti (University of Miami) Balinski’s theorem and Regularity of Line Arrangements

  22. Not all graphs are dual to a line arrangement Attention!, graphs like G 0 = { 12 , 34 } ∪ { 15 , 25 , 35 , 45 } ∪ { 16 , 26 , 36 , 46 } ∪ { 17 , 27 , 37 , 47 } are not dual to any Euclidean line arrangement! Try drawing it. Let P = r 1 ∩ r 2 and let Q = r 3 ∩ r 4 . Let p be the plane containing r 1 ∪ r 2 , and let q be the plane containing r 3 ∪ r 4 . How can a line meet all four r 1 , r 2 , r 3 , r 4 ? Bruno Benedetti (University of Miami) Balinski’s theorem and Regularity of Line Arrangements

  23. Not all graphs are dual to a line arrangement Attention!, graphs like G 0 = { 12 , 34 } ∪ { 15 , 25 , 35 , 45 } ∪ { 16 , 26 , 36 , 46 } ∪ { 17 , 27 , 37 , 47 } are not dual to any Euclidean line arrangement! Try drawing it. Let P = r 1 ∩ r 2 and let Q = r 3 ∩ r 4 . Let p be the plane containing r 1 ∪ r 2 , and let q be the plane containing r 3 ∪ r 4 . How can a line meet all four r 1 , r 2 , r 3 , r 4 ? There are only two chances (possibly coinciding): either it’s the line through P and Q , Bruno Benedetti (University of Miami) Balinski’s theorem and Regularity of Line Arrangements

  24. Not all graphs are dual to a line arrangement Attention!, graphs like G 0 = { 12 , 34 } ∪ { 15 , 25 , 35 , 45 } ∪ { 16 , 26 , 36 , 46 } ∪ { 17 , 27 , 37 , 47 } are not dual to any Euclidean line arrangement! Try drawing it. Let P = r 1 ∩ r 2 and let Q = r 3 ∩ r 4 . Let p be the plane containing r 1 ∪ r 2 , and let q be the plane containing r 3 ∪ r 4 . How can a line meet all four r 1 , r 2 , r 3 , r 4 ? There are only two chances (possibly coinciding): either it’s the line through P and Q , or it’s the line of intersection of the planes p ∩ q Bruno Benedetti (University of Miami) Balinski’s theorem and Regularity of Line Arrangements

  25. Not all graphs are dual to a line arrangement Attention!, graphs like G 0 = { 12 , 34 } ∪ { 15 , 25 , 35 , 45 } ∪ { 16 , 26 , 36 , 46 } ∪ { 17 , 27 , 37 , 47 } are not dual to any Euclidean line arrangement! Try drawing it. Let P = r 1 ∩ r 2 and let Q = r 3 ∩ r 4 . Let p be the plane containing r 1 ∪ r 2 , and let q be the plane containing r 3 ∪ r 4 . How can a line meet all four r 1 , r 2 , r 3 , r 4 ? There are only two chances (possibly coinciding): either it’s the line through P and Q , or it’s the line of intersection of the planes p ∩ q So two options!, not three. Bruno Benedetti (University of Miami) Balinski’s theorem and Regularity of Line Arrangements

  26. Not all graphs are dual to a line arrangement Attention!, graphs like G 0 = { 12 , 34 } ∪ { 15 , 25 , 35 , 45 } ∪ { 16 , 26 , 36 , 46 } ∪ { 17 , 27 , 37 , 47 } are not dual to any Euclidean line arrangement! Try drawing it. Let P = r 1 ∩ r 2 and let Q = r 3 ∩ r 4 . Let p be the plane containing r 1 ∪ r 2 , and let q be the plane containing r 3 ∪ r 4 . How can a line meet all four r 1 , r 2 , r 3 , r 4 ? There are only two chances (possibly coinciding): either it’s the line through P and Q , or it’s the line of intersection of the planes p ∩ q So two options!, not three. So some of the three lines r 5 , r 6 , r 7 have to coincide. Bruno Benedetti (University of Miami) Balinski’s theorem and Regularity of Line Arrangements

  27. Not all graphs are dual to a line arrangement Attention!, graphs like G 0 = { 12 , 34 } ∪ { 15 , 25 , 35 , 45 } ∪ { 16 , 26 , 36 , 46 } ∪ { 17 , 27 , 37 , 47 } are not dual to any Euclidean line arrangement! Try drawing it. Let P = r 1 ∩ r 2 and let Q = r 3 ∩ r 4 . Let p be the plane containing r 1 ∪ r 2 , and let q be the plane containing r 3 ∪ r 4 . How can a line meet all four r 1 , r 2 , r 3 , r 4 ? There are only two chances (possibly coinciding): either it’s the line through P and Q , or it’s the line of intersection of the planes p ∩ q So two options!, not three. So some of the three lines r 5 , r 6 , r 7 have to coincide. a contradiction Bruno Benedetti (University of Miami) Balinski’s theorem and Regularity of Line Arrangements

  28. Dual graphs of curves = all graphs Kollar 2012: every graph is dual to some arrangement of curves . Bruno Benedetti (University of Miami) Balinski’s theorem and Regularity of Line Arrangements

  29. Dual graphs of curves = all graphs Kollar 2012: every graph is dual to some arrangement of curves . Bruno Benedetti (University of Miami) Balinski’s theorem and Regularity of Line Arrangements

  30. Dual graphs of curves = all graphs Kollar 2012: every graph is dual to some arrangement of curves . IDEA: Start realizing K n with n random lines in P 2 ... Bruno Benedetti (University of Miami) Balinski’s theorem and Regularity of Line Arrangements

  31. Dual graphs of curves = all graphs Kollar 2012: every graph is dual to some arrangement of curves . IDEA: Start realizing K n with n random lines in P 2 ... Kyle Jenkins, Urban Geometry #296, acrilic on canvas, 2010 Bruno Benedetti (University of Miami) Balinski’s theorem and Regularity of Line Arrangements

  32. Dual graphs of curves = all graphs Kollar 2012: every graph is dual to some arrangement of curves . IDEA: Start realizing K n with n random lines in P 2 ... Kyle Jenkins, Urban Geometry #296, acrilic on canvas, 2010 Bruno Benedetti (University of Miami) Balinski’s theorem and Regularity of Line Arrangements

  33. Dual graphs of curves = all graphs Kollar 2012: every graph is dual to some arrangement of curves . IDEA: Start realizing K n with n random lines in P 2 ... Kyle Jenkins, Urban Geometry #296, acrilic on canvas, 2010 ...and then blowup “unwanted intersection points”. Bruno Benedetti (University of Miami) Balinski’s theorem and Regularity of Line Arrangements

  34. Dual graphs of curves = all graphs Kollar 2012: every graph is dual to some arrangement of curves . IDEA: Start realizing K n with n random lines in P 2 ... Kyle Jenkins, Urban Geometry #296, acrilic on canvas, 2010 ...and then blowup “unwanted intersection points”. So, Bruno Benedetti (University of Miami) Balinski’s theorem and Regularity of Line Arrangements

  35. Dual graphs of curves = all graphs Kollar 2012: every graph is dual to some arrangement of curves . IDEA: Start realizing K n with n random lines in P 2 ... Kyle Jenkins, Urban Geometry #296, acrilic on canvas, 2010 ...and then blowup “unwanted intersection points”. So, � dual graphs � dual graphs � dual graphs � � � � = = all graphs . of lines of curves of varieties Bruno Benedetti (University of Miami) Balinski’s theorem and Regularity of Line Arrangements

  36. Dual graphs of curves = all graphs Kollar 2012: every graph is dual to some arrangement of curves . IDEA: Start realizing K n with n random lines in P 2 ... Kyle Jenkins, Urban Geometry #296, acrilic on canvas, 2010 ...and then blowup “unwanted intersection points”. So, � dual graphs � dual graphs � dual graphs � � � � = = all graphs . of lines of curves of varieties It remains to see how dual graphs of simpl. complexes fit the hierarchy. Bruno Benedetti (University of Miami) Balinski’s theorem and Regularity of Line Arrangements

  37. Simplicial Complexes, Seen as Varieties (Stanley-Reisner) Definition by example: Bruno Benedetti (University of Miami) Balinski’s theorem and Regularity of Line Arrangements

  38. Simplicial Complexes, Seen as Varieties (Stanley-Reisner) Definition by example: Consider the simplicial complex ∆ below. Bruno Benedetti (University of Miami) Balinski’s theorem and Regularity of Line Arrangements

  39. Simplicial Complexes, Seen as Varieties (Stanley-Reisner) Definition by example: Consider the simplicial complex ∆ below. I ∆ := ( x 4 , x 5 , x 6 ) ∩ ( x 1 , x 5 , x 6 ) ∩ ( x 1 , x 2 , x 6 ) ∩ ( x 1 , x 2 , x 3 ) . Bruno Benedetti (University of Miami) Balinski’s theorem and Regularity of Line Arrangements

  40. Simplicial Complexes, Seen as Varieties (Stanley-Reisner) Definition by example: Consider the simplicial complex ∆ below. I ∆ := ( x 4 , x 5 , x 6 ) ∩ ( x 1 , x 5 , x 6 ) ∩ ( x 1 , x 2 , x 6 ) ∩ ( x 1 , x 2 , x 3 ) . (Prime ideals ↔ facets; each prime ideal just lists the variables corresponding to vertices that are not in that facet). Bruno Benedetti (University of Miami) Balinski’s theorem and Regularity of Line Arrangements

  41. Simplicial Complexes, Seen as Varieties (Stanley-Reisner) Definition by example: Consider the simplicial complex ∆ below. I ∆ := ( x 4 , x 5 , x 6 ) ∩ ( x 1 , x 5 , x 6 ) ∩ ( x 1 , x 2 , x 6 ) ∩ ( x 1 , x 2 , x 3 ) . (Prime ideals ↔ facets; each prime ideal just lists the variables corresponding to vertices that are not in that facet). � x 4 = 0 � x 1 = 0 � x 1 = 0 � x 1 = 0 � � � � V ( I ∆ ) = ∪ ∪ ∪ x 5 = 0 x 5 = 0 x 2 = 0 x 2 = 0 x 6 = 0 x 6 = 0 x 6 = 0 x 3 = 0 Bruno Benedetti (University of Miami) Balinski’s theorem and Regularity of Line Arrangements

  42. Simplicial Complexes, Seen as Varieties (Stanley-Reisner) Definition by example: Consider the simplicial complex ∆ below. I ∆ := ( x 4 , x 5 , x 6 ) ∩ ( x 1 , x 5 , x 6 ) ∩ ( x 1 , x 2 , x 6 ) ∩ ( x 1 , x 2 , x 3 ) . (Prime ideals ↔ facets; each prime ideal just lists the variables corresponding to vertices that are not in that facet). � x 4 = 0 � x 1 = 0 � x 1 = 0 � x 1 = 0 � � � � V ( I ∆ ) = ∪ ∪ ∪ x 5 = 0 x 5 = 0 x 2 = 0 x 2 = 0 x 6 = 0 x 6 = 0 x 6 = 0 x 3 = 0 Dual graph of V ( I ∆ )? Bruno Benedetti (University of Miami) Balinski’s theorem and Regularity of Line Arrangements

  43. Simplicial Complexes, Seen as Varieties (Stanley-Reisner) Definition by example: Consider the simplicial complex ∆ below. I ∆ := ( x 4 , x 5 , x 6 ) ∩ ( x 1 , x 5 , x 6 ) ∩ ( x 1 , x 2 , x 6 ) ∩ ( x 1 , x 2 , x 3 ) . (Prime ideals ↔ facets; each prime ideal just lists the variables corresponding to vertices that are not in that facet). � x 4 = 0 � x 1 = 0 � x 1 = 0 � x 1 = 0 � � � � V ( I ∆ ) = ∪ ∪ ∪ x 5 = 0 x 5 = 0 x 2 = 0 x 2 = 0 x 6 = 0 x 6 = 0 x 6 = 0 x 3 = 0 Dual graph of V ( I ∆ )? The intersection of the first 2 components is { x : x 4 = x 5 = x 6 = x 1 = 0 } , Bruno Benedetti (University of Miami) Balinski’s theorem and Regularity of Line Arrangements

  44. Simplicial Complexes, Seen as Varieties (Stanley-Reisner) Definition by example: Consider the simplicial complex ∆ below. I ∆ := ( x 4 , x 5 , x 6 ) ∩ ( x 1 , x 5 , x 6 ) ∩ ( x 1 , x 2 , x 6 ) ∩ ( x 1 , x 2 , x 3 ) . (Prime ideals ↔ facets; each prime ideal just lists the variables corresponding to vertices that are not in that facet). � x 4 = 0 � x 1 = 0 � x 1 = 0 � x 1 = 0 � � � � V ( I ∆ ) = ∪ ∪ ∪ x 5 = 0 x 5 = 0 x 2 = 0 x 2 = 0 x 6 = 0 x 6 = 0 x 6 = 0 x 3 = 0 Dual graph of V ( I ∆ )? The intersection of the first 2 components is { x : x 4 = x 5 = x 6 = x 1 = 0 } , which is 2-dimensional Bruno Benedetti (University of Miami) Balinski’s theorem and Regularity of Line Arrangements

  45. Simplicial Complexes, Seen as Varieties (Stanley-Reisner) Definition by example: Consider the simplicial complex ∆ below. I ∆ := ( x 4 , x 5 , x 6 ) ∩ ( x 1 , x 5 , x 6 ) ∩ ( x 1 , x 2 , x 6 ) ∩ ( x 1 , x 2 , x 3 ) . (Prime ideals ↔ facets; each prime ideal just lists the variables corresponding to vertices that are not in that facet). � x 4 = 0 � x 1 = 0 � x 1 = 0 � x 1 = 0 � � � � V ( I ∆ ) = ∪ ∪ ∪ x 5 = 0 x 5 = 0 x 2 = 0 x 2 = 0 x 6 = 0 x 6 = 0 x 6 = 0 x 3 = 0 Dual graph of V ( I ∆ )? The intersection of the first 2 components is { x : x 4 = x 5 = x 6 = x 1 = 0 } , which is 2-dimensional ⇒ edge! Bruno Benedetti (University of Miami) Balinski’s theorem and Regularity of Line Arrangements

  46. Simplicial Complexes, Seen as Varieties (Stanley-Reisner) Definition by example: Consider the simplicial complex ∆ below. I ∆ := ( x 4 , x 5 , x 6 ) ∩ ( x 1 , x 5 , x 6 ) ∩ ( x 1 , x 2 , x 6 ) ∩ ( x 1 , x 2 , x 3 ) . (Prime ideals ↔ facets; each prime ideal just lists the variables corresponding to vertices that are not in that facet). � x 4 = 0 � x 1 = 0 � x 1 = 0 � x 1 = 0 � � � � V ( I ∆ ) = ∪ ∪ ∪ x 5 = 0 x 5 = 0 x 2 = 0 x 2 = 0 x 6 = 0 x 6 = 0 x 6 = 0 x 3 = 0 Dual graph of V ( I ∆ )? The intersection of the first 2 components is { x : x 4 = x 5 = x 6 = x 1 = 0 } , which is 2-dimensional ⇒ edge! The intersection of the first and third component is { x : x 4 = x 5 = x 6 = x 1 = x 2 = 0 } , Bruno Benedetti (University of Miami) Balinski’s theorem and Regularity of Line Arrangements

  47. Simplicial Complexes, Seen as Varieties (Stanley-Reisner) Definition by example: Consider the simplicial complex ∆ below. I ∆ := ( x 4 , x 5 , x 6 ) ∩ ( x 1 , x 5 , x 6 ) ∩ ( x 1 , x 2 , x 6 ) ∩ ( x 1 , x 2 , x 3 ) . (Prime ideals ↔ facets; each prime ideal just lists the variables corresponding to vertices that are not in that facet). � x 4 = 0 � x 1 = 0 � x 1 = 0 � x 1 = 0 � � � � V ( I ∆ ) = ∪ ∪ ∪ x 5 = 0 x 5 = 0 x 2 = 0 x 2 = 0 x 6 = 0 x 6 = 0 x 6 = 0 x 3 = 0 Dual graph of V ( I ∆ )? The intersection of the first 2 components is { x : x 4 = x 5 = x 6 = x 1 = 0 } , which is 2-dimensional ⇒ edge! The intersection of the first and third component is { x : x 4 = x 5 = x 6 = x 1 = x 2 = 0 } , which is 1-dim. Bruno Benedetti (University of Miami) Balinski’s theorem and Regularity of Line Arrangements

  48. Simplicial Complexes, Seen as Varieties (Stanley-Reisner) Definition by example: Consider the simplicial complex ∆ below. I ∆ := ( x 4 , x 5 , x 6 ) ∩ ( x 1 , x 5 , x 6 ) ∩ ( x 1 , x 2 , x 6 ) ∩ ( x 1 , x 2 , x 3 ) . (Prime ideals ↔ facets; each prime ideal just lists the variables corresponding to vertices that are not in that facet). � x 4 = 0 � x 1 = 0 � x 1 = 0 � x 1 = 0 � � � � V ( I ∆ ) = ∪ ∪ ∪ x 5 = 0 x 5 = 0 x 2 = 0 x 2 = 0 x 6 = 0 x 6 = 0 x 6 = 0 x 3 = 0 Dual graph of V ( I ∆ )? The intersection of the first 2 components is { x : x 4 = x 5 = x 6 = x 1 = 0 } , which is 2-dimensional ⇒ edge! The intersection of the first and third component is { x : x 4 = x 5 = x 6 = x 1 = x 2 = 0 } , which is 1-dim. ⇒ no edge! Bruno Benedetti (University of Miami) Balinski’s theorem and Regularity of Line Arrangements

  49. Simplicial Complexes, Seen as Varieties (Stanley-Reisner) Definition by example: Consider the simplicial complex ∆ below. I ∆ := ( x 4 , x 5 , x 6 ) ∩ ( x 1 , x 5 , x 6 ) ∩ ( x 1 , x 2 , x 6 ) ∩ ( x 1 , x 2 , x 3 ) . (Prime ideals ↔ facets; each prime ideal just lists the variables corresponding to vertices that are not in that facet). � x 4 = 0 � x 1 = 0 � x 1 = 0 � x 1 = 0 � � � � V ( I ∆ ) = ∪ ∪ ∪ x 5 = 0 x 5 = 0 x 2 = 0 x 2 = 0 x 6 = 0 x 6 = 0 x 6 = 0 x 3 = 0 Dual graph of V ( I ∆ )? The intersection of the first 2 components is { x : x 4 = x 5 = x 6 = x 1 = 0 } , which is 2-dimensional ⇒ edge! The intersection of the first and third component is { x : x 4 = x 5 = x 6 = x 1 = x 2 = 0 } , which is 1-dim. ⇒ no edge! ... So dual graph of V ( I ∆ ) is same of ∆. Bruno Benedetti (University of Miami) Balinski’s theorem and Regularity of Line Arrangements

  50. { dual graphs of complexes } ⊂ { dual graphs of lines } Bruno Benedetti (University of Miami) Balinski’s theorem and Regularity of Line Arrangements

  51. { dual graphs of complexes } ⊂ { dual graphs of lines } Stanley-Reisner : simplicial complexes on n vertices are in bijection with radical monomial ideals in C [ x 1 , . . . , x n ]. Bruno Benedetti (University of Miami) Balinski’s theorem and Regularity of Line Arrangements

  52. { dual graphs of complexes } ⊂ { dual graphs of lines } Stanley-Reisner : simplicial complexes on n vertices are in bijection with radical monomial ideals in C [ x 1 , . . . , x n ]. Zariski : radical ideals I in C [ x 1 , . . . , x n ] are in bijection with algebraic objects V ( I ) in A n . Bruno Benedetti (University of Miami) Balinski’s theorem and Regularity of Line Arrangements

  53. { dual graphs of complexes } ⊂ { dual graphs of lines } Stanley-Reisner : simplicial complexes on n vertices are in bijection with radical monomial ideals in C [ x 1 , . . . , x n ]. Zariski : radical ideals I in C [ x 1 , . . . , x n ] are in bijection with algebraic objects V ( I ) in A n . Composing the two, from any complex ∆ we get an algebraic object V ( I ∆ ) ⊂ A n . Bruno Benedetti (University of Miami) Balinski’s theorem and Regularity of Line Arrangements

  54. { dual graphs of complexes } ⊂ { dual graphs of lines } Stanley-Reisner : simplicial complexes on n vertices are in bijection with radical monomial ideals in C [ x 1 , . . . , x n ]. Zariski : radical ideals I in C [ x 1 , . . . , x n ] are in bijection with algebraic objects V ( I ) in A n . Composing the two, from any complex ∆ we get an algebraic object V ( I ∆ ) ⊂ A n . A special variety (a coordinate subspace arrangement): Bruno Benedetti (University of Miami) Balinski’s theorem and Regularity of Line Arrangements

  55. { dual graphs of complexes } ⊂ { dual graphs of lines } Stanley-Reisner : simplicial complexes on n vertices are in bijection with radical monomial ideals in C [ x 1 , . . . , x n ]. Zariski : radical ideals I in C [ x 1 , . . . , x n ] are in bijection with algebraic objects V ( I ) in A n . Composing the two, from any complex ∆ we get an algebraic object V ( I ∆ ) ⊂ A n . A special variety (a coordinate subspace arrangement): So when we do generic hyperplane sections, we get an arrangement of lines . Bruno Benedetti (University of Miami) Balinski’s theorem and Regularity of Line Arrangements

  56. { dual graphs of complexes } ⊂ { dual graphs of lines } Stanley-Reisner : simplicial complexes on n vertices are in bijection with radical monomial ideals in C [ x 1 , . . . , x n ]. Zariski : radical ideals I in C [ x 1 , . . . , x n ] are in bijection with algebraic objects V ( I ) in A n . Composing the two, from any complex ∆ we get an algebraic object V ( I ∆ ) ⊂ A n . A special variety (a coordinate subspace arrangement): So when we do generic hyperplane sections, we get an arrangement of lines . FACT For any simplicial complex ∆, the dual graphs of ∆ and of V ( I ∆ ) are the same. Bruno Benedetti (University of Miami) Balinski’s theorem and Regularity of Line Arrangements

  57. { dual graphs of complexes } ⊂ { dual graphs of lines } Stanley-Reisner : simplicial complexes on n vertices are in bijection with radical monomial ideals in C [ x 1 , . . . , x n ]. Zariski : radical ideals I in C [ x 1 , . . . , x n ] are in bijection with algebraic objects V ( I ) in A n . Composing the two, from any complex ∆ we get an algebraic object V ( I ∆ ) ⊂ A n . A special variety (a coordinate subspace arrangement): So when we do generic hyperplane sections, we get an arrangement of lines . FACT For any simplicial complex ∆, the dual graphs of ∆ and of V ( I ∆ ) are the same. Bruno Benedetti (University of Miami) Balinski’s theorem and Regularity of Line Arrangements

  58. { dual graphs of complexes } ⊂ { dual graphs of lines } Stanley-Reisner : simplicial complexes on n vertices are in bijection with radical monomial ideals in C [ x 1 , . . . , x n ]. Zariski : radical ideals I in C [ x 1 , . . . , x n ] are in bijection with algebraic objects V ( I ) in A n . Composing the two, from any complex ∆ we get an algebraic object V ( I ∆ ) ⊂ A n . A special variety (a coordinate subspace arrangement): So when we do generic hyperplane sections, we get an arrangement of lines . FACT For any simplicial complex ∆, the dual graphs of ∆ and of V ( I ∆ ) are the same. � dual graphs � � � dual graphs of ⊂ . This implies simplicial complexes of lines (Graphs like { 12 , 13 , 15 , 23 , 24 , 34 , 45 } show the containment is strict.) Bruno Benedetti (University of Miami) Balinski’s theorem and Regularity of Line Arrangements

  59. Conclusions of Part I. Bruno Benedetti (University of Miami) Balinski’s theorem and Regularity of Line Arrangements

  60. Conclusions of Part I. The notion of “dual graph” can be lifted from simplicial complexes to algebraic varieties. Bruno Benedetti (University of Miami) Balinski’s theorem and Regularity of Line Arrangements

  61. Conclusions of Part I. The notion of “dual graph” can be lifted from simplicial complexes to algebraic varieties. (We can restrict ourselves to dimension one if you wish, so curves or lines.) Bruno Benedetti (University of Miami) Balinski’s theorem and Regularity of Line Arrangements

  62. Conclusions of Part I. The notion of “dual graph” can be lifted from simplicial complexes to algebraic varieties. (We can restrict ourselves to dimension one if you wish, so curves or lines.) Statements on graphs of polytopes (like Balinski’s theorem, or diameter bounds), might extend to this more general world: Bruno Benedetti (University of Miami) Balinski’s theorem and Regularity of Line Arrangements

  63. Conclusions of Part I. The notion of “dual graph” can be lifted from simplicial complexes to algebraic varieties. (We can restrict ourselves to dimension one if you wish, so curves or lines.) Statements on graphs of polytopes (like Balinski’s theorem, or diameter bounds), might extend to this more general world: Bruno Benedetti (University of Miami) Balinski’s theorem and Regularity of Line Arrangements

  64. Conclusions of Part I. The notion of “dual graph” can be lifted from simplicial complexes to algebraic varieties. (We can restrict ourselves to dimension one if you wish, so curves or lines.) Statements on graphs of polytopes (like Balinski’s theorem, or diameter bounds), might extend to this more general world: Example Bruno Benedetti (University of Miami) Balinski’s theorem and Regularity of Line Arrangements

  65. Conclusions of Part I. The notion of “dual graph” can be lifted from simplicial complexes to algebraic varieties. (We can restrict ourselves to dimension one if you wish, so curves or lines.) Statements on graphs of polytopes (like Balinski’s theorem, or diameter bounds), might extend to this more general world: Example (from 3 slides forward - ignore obscure words for now) For any ( d − 1)-sphere ∆, the variety V ( I ∆ ) is an arithmetically Gorenstein subspace arrangement of Castelnuovo–Mumford regularity d + 1. Bruno Benedetti (University of Miami) Balinski’s theorem and Regularity of Line Arrangements

  66. Conclusions of Part I. The notion of “dual graph” can be lifted from simplicial complexes to algebraic varieties. (We can restrict ourselves to dimension one if you wish, so curves or lines.) Statements on graphs of polytopes (like Balinski’s theorem, or diameter bounds), might extend to this more general world: Example (from 3 slides forward - ignore obscure words for now) For any ( d − 1)-sphere ∆, the variety V ( I ∆ ) is an arithmetically Gorenstein subspace arrangement of Castelnuovo–Mumford regularity d + 1. Bruno Benedetti (University of Miami) Balinski’s theorem and Regularity of Line Arrangements

  67. Conclusions of Part I. The notion of “dual graph” can be lifted from simplicial complexes to algebraic varieties. (We can restrict ourselves to dimension one if you wish, so curves or lines.) Statements on graphs of polytopes (like Balinski’s theorem, or diameter bounds), might extend to this more general world: Example (from 3 slides forward - ignore obscure words for now) For any ( d − 1)-sphere ∆, the variety V ( I ∆ ) is an arithmetically Gorenstein subspace arrangement of Castelnuovo–Mumford regularity d + 1. Maybe elementary facts like “the dual graph of any ( d − 1)-sphere ∆ is d -connected” (Klee-Balinski) can be proven with algebraic methods? Bruno Benedetti (University of Miami) Balinski’s theorem and Regularity of Line Arrangements

  68. Conclusions of Part I. The notion of “dual graph” can be lifted from simplicial complexes to algebraic varieties. (We can restrict ourselves to dimension one if you wish, so curves or lines.) Statements on graphs of polytopes (like Balinski’s theorem, or diameter bounds), might extend to this more general world: Example (from 3 slides forward - ignore obscure words for now) For any ( d − 1)-sphere ∆, the variety V ( I ∆ ) is an arithmetically Gorenstein subspace arrangement of Castelnuovo–Mumford regularity d + 1. Maybe elementary facts like “the dual graph of any ( d − 1)-sphere ∆ is d -connected” (Klee-Balinski) can be proven with algebraic methods? Bruno Benedetti (University of Miami) Balinski’s theorem and Regularity of Line Arrangements

  69. Part II. The Algebraic Machinery (sketch). Bruno Benedetti (University of Miami) Balinski’s theorem and Regularity of Line Arrangements

  70. Complete intersections Bruno Benedetti (University of Miami) Balinski’s theorem and Regularity of Line Arrangements

  71. Complete intersections Linear algebra : every k -dimensional subspace X of P n can be described with exactly n − k linear equations. Bruno Benedetti (University of Miami) Balinski’s theorem and Regularity of Line Arrangements

  72. Complete intersections Linear algebra : every k -dimensional subspace X of P n can be described with exactly n − k linear equations. Non-Linear algebra : The best we can say about a variety X ⊂ P n , is that we need at least n − k (polynomial) equations. Bruno Benedetti (University of Miami) Balinski’s theorem and Regularity of Line Arrangements

  73. Complete intersections Linear algebra : every k -dimensional subspace X of P n can be described with exactly n − k linear equations. Non-Linear algebra : The best we can say about a variety X ⊂ P n , is that we need at least n − k (polynomial) equations. Bruno Benedetti (University of Miami) Balinski’s theorem and Regularity of Line Arrangements

  74. Complete intersections Linear algebra : every k -dimensional subspace X of P n can be described with exactly n − k linear equations. Non-Linear algebra : The best we can say about a variety X ⊂ P n , is that we need at least n − k (polynomial) equations. Complete intersections are the varieties for which “=” holds. The ”twisted cubic” ( s 3 , s 2 t , st 2 , t 3 ) of P 3 is not a complete intersection: one needs at least three (hyper)surfaces to cut it out. Bruno Benedetti (University of Miami) Balinski’s theorem and Regularity of Line Arrangements

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