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Jul 28, 2010 at 19:36 comment added Tony Scholl If you believe the Tate conjectures, then every motive belongs to the tensor category generated by motives of curves. I recall from somewhere that Grothendieck had the idea to prove the Weil conjectures by covering varieties with products of curves; then Serre found a surface that couldn't be covered rationally by a product of curves (see Grothendieck-Serre correspondence, 31/3/1964).
Jul 28, 2010 at 16:55 history answered Donu Arapura CC BY-SA 2.5