In two different situations we describe fibrations via divisors on the base space with rather unusual coefficients: First, branched coverings X -> P1 that are the quotient of the action of a finite group A can be encoded by a degree zero divisor on P1 with coefficients in A. Second, degenerate toric fibrations X -> Y with generic fiber F = TV(sigma) (the toric variety associated to a cone sigma) correspond to a divisor on Y with coefficients being polyhedra in the space N of one-parameter subgroups of the torus.
We report on a number of computer experiments, joint work in progress with Alessio Corti, Sergei Galkin, Vasily Golyshev, Al Kasprzyk, concerning Laurent polynomials of low ramification and Fano 3-folds.
In this talk we will introduce the notion of gaps of spectra of categories and we will consider some applications to birational geometry and theory of algebraic cycles. Examples will be considered.
Among all type $D$ Grassmannians $O(m,2n)$, the minuscule ones ($m=1$ and $m=n$) have quantum cohomology rings that are easiest to understand. We recall the basic structure results (ring presentation, Pieri and Giambelli formulas) and contrast them with the corresponding results for non-minuscule type $D$ Grassmannians as well as (co)minuscule homogeneous varieties of other Lie types.
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