The sessions of the sections are highlighted in blue, those of the mini-symposia in yellow.
Please select a date or location to show only sessions at that day or location. If you click the selected day again, you return to the agenda overview.
You can also filter by sections or mini-symposia (topics).
Please select a single session for detailed view with abstracts.
As participant you can create your own personal agenda. To do so, log into your account first. Then go to the agenda and click on the plus symbol to add sessions to your personal agenda.
Toric manifolds of small Picard number: computer assisted classification
Suyoung Choi1, Jang Hyeontae2, Mathieu Vallée3
1: Ajou University, Korea; 2: Korea Institute for Advanced Studies, Korea; 3: Université Libre de Bruxelles, Belgium
Toric varieties are algebraic varieties equipped with a well-behaved torus action, admitting an explicit combinatorial description. The fundamental theorem of toric geometry establishes a correspondence between toric varieties and fans: collections of strongly convex polyhedral cones in $\mathbb{R}^n$, closed under taking faces and with pairwise disjoint relative interiors. Fan properties translate directly into geometry: a toric variety is complete if and only if its cones cover $\mathbb{R}^n$, and non-singular if and only if each cone is generated by part of a $\mathbb{Z}^n$-basis. We focus on complete non-singular toric varieties (toric manifolds). The Picard number equals the number of 1-dimensional cones minus $n$.
Two research directions arise: fixed dimension or fixed Picard number. In dimension 2, toric manifolds are fully classified via toric blow-ups of $\mathbb{CP}^2$ or Hirzebruch surfaces. In any dimension $n$, the unique toric manifold of Picard number 1 is $\mathbb{CP}^n$, whose fan is the normal fan of a unimodular $n$-simplex. Kleinschmidt (1988) and Batyrev (1991) classified Picard numbers 2 and 3 respectively.
We present joint work with Choi and Jang classifying toric manifolds of Picard number 4, via the simplicial wedge operation --- notably used by Santos to disprove the Hirsch conjecture --- together with a computer assisted approach. Time permitting, we discuss a dynamic programming approach toward Picard number 5.
