From Notices of the AMS
Curve-Counting and Mirror Symmetry
by Emily Clader
Communicated by Han-Bom Moon
Curve-counting is a subject that dates back hundreds, even thousands, of years. Broadly speaking, its goal is to answer questions about the number of curves in some ambient space that satisfy prescribed conditions, such as the following:
How many conics pass through five given points in the plane?
How many lines pass through four given lines in three-space?
How many lines lie on the quintic threefold \[\{z_1^5 + z_2^5 + \cdots + z_5^5 = 0\}\] in ℂ ℙ4?
The answer to the first of these questions was known to the ancient Greeks: given five (sufficiently general) points in ℝ2, there is exactly one conic that passes through all five of them. The method by which the ancient Greeks would have arrived at this answer is by an explicit construction, given the coordinates of the five points, of the conic that passes through them.
- Also in Notices
- Equiangular Lines and Eigenvalue Multiplicities
- Rate-Based Synaptic Plasticity Rules
