Short Summer School
The derived category of cubic hypersurfaces

All the lectures will take place at the historical building of the University of Barcelona in the city center, room B5.

July 18th	9:30	Venue	Room B5, Faculty of Math., Gran Via 585
	10-11	M. Lahoz	Semi-orthogonal decompositions, HPD, Quadric fibrations - I
	11	Coffee-break
	11:30-12:30	M. Lahoz	Semi-orthogonal decompositions, HPD, Quadric fibrations - II
	15-16	E. Macrì	Stability conditions: the case of curves and basic properties
July 19th	10-11	P. Stellari	3-folds: Hodge theory and derived categories
	11	Coffee-break
	11:30-12:30	M. Lahoz	4-folds: geometry
	15-16	E. Macrì	Stability conditions in higher dimension: surfaces
July 20th	9-10	P. Stellari	4-folds: derived categories, Fano varieties, Addington-Thomas, ACM
	10 Coffee-break
	10:30-11:30	E. Macrì	Gaps and rationality
	12-13	P. Stellari	Matrix factorization and the Hodge conjecture

Contents

In this short summer school we would like to present the derived category new approach to the rationality problem of cubic threefolds and fourfolds and compare it with the classical Hodge theoretical approach. Short summary of contents:

• Preliminaries about the derived category and semi-orthogonal decompositions;
• Stability conditions on the derived category;
• Cubic threefolds. Hodge theory and derived category;
• Hodge theoretical results for cubic fourfolds;
• Derived category of cubic fourfolds;
• Rationality conjectures in the derived category.

(pdf)