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Applications of toric systems on surfaces. (English) Zbl 1403.14050

Orlov’s folklore conjecture states that a smooth projective surface admitting a full exceptional collection must be a rational surface. In this paper, the author, using the techniques of toric systems introduced by L. Hille and M. Perling [Compos. Math. 147, No. 4, 1230–1280 (2011; Zbl 1237.14043)], proves some partial results on this conjecture: he proves that a smooth projective surface with a strong exceptional collection of line bundles of maximal length should be rational (Theorem 1.2) (improving a result by M. Brown and I. Shipman [Mich. Math. J. 66, No. 4, 785–811 (2017; Zbl 1429.14013)]).
The second main result deals with Orlov’s conjecture for surfaces with small Picard number: he proves that a smooth projective surface \(X\) with Picard number \(\rho(X)\leq 3\) admitting a full exceptional collection is rational (Theorem 1.4).
Finally, he gives a partial solution to a conjecture by S. Okawa and H. Uehara [Int. Math. Res. Not. 2015, No. 23, 12781–12803 (2015; Zbl 1376.13007)] about exceptional sheaves on weak del Pezzo surfaces (Theorem 1.13).

MSC:

14F05 Sheaves, derived categories of sheaves, etc. (MSC2010)
14J26 Rational and ruled surfaces
14M25 Toric varieties, Newton polyhedra, Okounkov bodies
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References:

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