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Regulus on a Hyperboloid
A quadric surface (for example, a hyperboloid) supports two reguli: a family of disjoint lines that cover the quadric surface!
A quadric surface is an example of a ruled surface, meaning it admits a map π to some curve C (in this case, C could be a quadric curve or a line itself) where the fibers are each ℙ1, and π admits a section from C back to the surface. The "Mysterious Configuration" This is a (10,12)-geproci configuration of 120 points in ℙ3ℂ, as explored in this paper by Chiantini, Farnik, Favacchio, Harbourne, Migliore, Szemberg, and Szpond. All the points of this configuration are real, but 21 are at the plane at infinity, so only 99 are visible. This is a (1204,806)-configuration, but 4 of the 80 lines are in the plane at infinity, so only 76 are visible. I also made a version with only the points, and a version showing a spread of the configuration (that is, a covering the set of points with by disjoint lines). Another Extremal Elliptic Surface This is a surface induced by the elliptic fibration spanned by the polynomials x3+y3+z3 and xyz-y2z. Its Mordell-Weil group is isomorphic to ℤ6 x ℤ6, but only two of its 36 sections are visible. It has one special fiber of Kodaira type B3, and nine of type I1. A Non-Extremal Elliptic Surface with Double Points This is a surface induced by the elliptic fibration spanned by the polynomials x3-xz2 and (y-z)2(y+z). It is non-extremal: its Mordell-Weil group is infinite. The three red lines are the sections corresponding to the exceptional divisors of the blowup, and the three orange lines are (-2)-curves. Its Dynkin diagram is D4 + A2. |
12 Lines on the Clebsch Cubic
A smooth cubic surface in ℙ3ℂ will have 27 lines, but not all of them will be real! This portrays the 12 real lines of the Clebsch cubic surface in 𝔸3ℝ defined by the polynomial x3+y3+z3+1=(x+y+z+1)3. The Clebsch surface is special because it contains the maximal number of Eckardt points (points where three lines on the surface meet) of 10. Although, only four of them are real.
The Hesse Pencil The Hesse pencil is the span of the polynomials x3+y3+z3 and its Hesse derivative xyz in ℙ2. We can realize this pencil as a surface (in blue) in three-dimensional space by allowing the parameter to control the height. In silver, we see the fibers of the pencil, which are different linear combinations of x3+y3+z3 and xyz. This elliptic surface is extremal, and the Mordell-Weil group is isomorphic to ℤ3 x ℤ3. Only two of its nine sections are visible here (in red). Its Dynkin diagram is 4Ã2. A Non-Extremal Elliptic Surface This is a surface induced by the elliptic fibration spanned by the polynomials x3-xz2 and y3-yz2. It is non-extremal: its Mordell-Weil group is infinite. The nine red lines are the sections corresponding to the exceptional divisors of the blowup, the four orange curves are examples of other sections in the Mordell-Weil group. Its Dynkin diagram is 2Ã1 + 2Ã2. |