Geometry & Topology Seminar

Toric Systems on Weak del Pezzo surfaces

abstract: After recalling  the definition of (strong) exceptional collection of line bundles on smooth projective surface X, I will introduce a technical tool called toric system coming from exceptional collection invented by L.Hille and M.Perling and various operations on it. Hille and Perling conjectured that all the full strong exceptional toric system is coming from augmentation(an analogue operations of blow up of a point) from the ones on P^2 and Hirzebruch surfaces. I will show that the conjecture holds for cyclic strong exceptional toric systems on any rational surface. As a result, I will show that the existence of cyclic strong exceptional toric systems will imply the rationality of smooth projective surfaces and the anticanonical divisor -K_X is big and nef. Conversely, I will give a complete classification of weak del pezzo surfaces which admits cyclic strong exceptional collection of line bundles. Next, I will sketch the idea of proof of Hille-Perling's conjecture for weak del Pezzo surfaces of K_X^2\geq 3. Then, I will give a counter example of the conjecture on a weak del Pezzo surface of degree 2. If time permits, I will talk about the application of toric systems in several conjectures of D.Orlov on dimension of $D^b(coh X)$ for smooth variety and rationality of surfaces admitting a full exceptional collection of line bundles. This is a joint work with Alexey Elagin