On the derived category of the classical Godeaux surface

Abstract

We construct an exceptional sequence of length 11 on the classical Godeaux surface X which is the Z/5-quotient of the Fermat quintic surface in P^3. This is the maximal possible length of such a sequence on this surface which has Grothendieck group Z^11+Z/5. In particular, the result answers Kuznetsov's Nonvanishing Conjecture, which concerns Hochschild homology of an admissible subcategory, in the negative. The sequence carries a symmetry when interpreted in terms of the root lattice of the simple Lie algebra of type E_8. We also produce explicit nonzero objects in the (right) orthogonal to the exceptional sequence.Comment: 33 pages, 1 figure; version 2: apart from small changes, section 10 on the derived endomorphism algebra of the sequence adde

Exceptional collections, and the Néron–Severi lattice for surfaces

“…Proof of Claim 2. For this purpose, given Claim 1, we show that a characteristic vector ξ := x 1 e 1 + · · · + x n e n of norm b(ξ, ξ) = 10 − n and whose coordinates satisfy (7) is necessarily the vector 3e 1 + e 2 + · · · + e n . Squaring the inequality x 4 + x 3 + x 2 ≤ x 1 yields (8) 2(x 2 x 3 + x 2 x 4 + x 3 x 4 ) ≤ x 2 1 − x 2 2 − x 2 3 − x 2 4 = 10 − n + x 2 5 + · · · + x 2 n .…”

“…Let ω be an element of Λ such that b(ω, ω) ≥ 0. Then there is an automorphism ϕ of Λ preserving q (i.e., ϕ ∈ O(q)) such that ϕ(ω) = x 1 e 1 + · · · + x n e n with (7) 0 ≤ x n ≤ x n−1 ≤ · · · ≤ x 1 and x 4 + x 3 + x 2 ≤ x 1 .…”

“…Numerical constraints. Although the problem of classifying smooth projective complex surfaces that admit a full exceptional collection seems out of reach at present (it is conjectured that only the surfaces that are rational have a full exceptional collection), a fair amount of work [1,7,8,16] has been carried out in order to construct exceptional collections of maximal length on complex surfaces with p g = q = 0. (As usual, for a smooth projective surface S, the geometric genus is p g := h 0 (Ω 2 S ) = h 2 (O S ) and the irregularity is q := h 1 (O S ).)…”

Birational Geometry via Moduli Spaces

“…4. Following [DKK12b] and the pioneering work [BBS12] we develop the notions of phantom category and we emphasize its connection with introduced in this paper notion of a moving scheme. The last one determines the geometry of the moduli space of LG models and as a result the geometry of the initial manifold.…”

Applications of toric systems on surfaces

“…An immediate corollary of this theorem is that a surface admitting a cyclic strong exceptional collection of line bundles of maximal length must be a rational surface. An alternating proof of this fact is given in [12,Lemma 15.1] Motivated by the desire to describe geometric phantom categories: subcategories of D b (X) with trivial Grothendieck group K 0 and Hochschild homology HH 0 , a large amount of work was carried out to establish the exceptional collection of line bundles of maximal length on a surface of general type with p g = q = 0, see [1], [4], [13], [14]. It immediately follows from Theorem 1.2 that: Corollary 1.3.…”