Kahn-Sujatha's birational motive is a variant of Chow motive that synthesis the ideas of birational geometry and motives. We explain our result saying that the unramified cohomology is a universal invariant for torsion motives of surfaces. We also exhibit examples of complex varieties violating the integral Hodge conjecture. If time permits, we discuss a pathology in positive characteristic. (Joint work with Kanetomo Sato.)

For each positive integer q del Pezzo q-secant varieties are subextremal objects, in a natural sense, among q-secant varieties to nondegenerate projective varieties. In this talk we review their definition, properties, and examples, together with those of extremal objects, namely q-secant varieties of minimal degree.