Mirror symmetry and the classification of orbifold del Pezzo surfaces

Abstract

We state a number of conjectures that together allow one to classify a broad class of del Pezzo surfaces with cyclic quotient singularities using mirror symmetry. We prove our conjectures in the simplest cases. The conjectures relate mutation-equivalence classes of Fano polygons with Q-Gorenstein deformation classes of del Pezzo surfaces.where x, y, z are the standard co-ordinate functions on C 3 . Write w = mr + w 0 with 0 ≤ w 0 < r. It is known [24,25] that the base of the miniversal qG-deformation 2 of 1 n (1, q) is isomorphic to C m−1 and, choosing co-ordinate functions a 1 , . . . , a m−1 on it, the miniversal qG-family is given explicitly by the equation:xy + (z rm + a 1 z r(m−2) + · · · + a m−1 )z w0 = 0 ⊂ 1 r (1, w 0 a − 1, a) × C m−1 We say that 1 n (1, q) is of class T or is a T -singularity if w 0 = 0, and that it is a primitive T -singularity if w 0 = 0 and m = 1. T -singularities appear in the work of Wahl [28] and Kollár-Shepherd-Barron [25]. We say that 1 n (1, q) is of class R or is a residual singularity if m = 0, that is, if w = w 0 . We say that the singularity

Minimality and Mutation-Equivalence Of polygons

“…In the cases (5, 5, 2), (8,4,1), and (9, 3, 1), we have equality in (9), hence 1 = Vol(P * ) = Vol(T * ) and so P = T is uniquely determined. This gives the triangles T 1 , T 4 , and T 9 .…”

“…Let w = (0, −1) ∈ M, so that h min = −1 and h max = 2, and set F = conv{0, (1, 0)} ⊂ N Q . Then F is a factor of P (1,1,1) with respect to w, giving the mutation P (1,1,2) := mut w (P (1,1,1) , F) with vertices (0, 1), (−1, −2), (1, −2) as depicted below. The toric variety corresponding to P (1,1,2) is P(1, 1, 4).…”

“…We can draw a graph of all possible mutations obtainable from P (1,1,1) : the vertices of the graph denote GL 2 (Z)-equivalence classes of Fano polygons, and two vertices are connected by an edge if there exists a mutation between the two Fano polygons (notice that, since mutations are invertible, we can regard the edges as being undirected). We obtain a tree whose typical vertex is trivalent [4, Example 3.14]:…”

“…As noted in Section 1.1, the toric variety X P is qG-deformation-equivalent to a del Pezzo surface X with singular points B and topological Euler number of X \Sing(X ) equal to n [1,3]. The degree and Hilbert series can be expressed purely in terms of singularity content.…”

“…Ilten [26] has shown that if two Fano polytopes P and Q are related by mutation then the corresponding toric varieties X P and X Q are deformation equivalent: there exists a flat family X → P 1 such that X 0 ∼ = X P and X ∞ ∼ = X Q . In fact X P and X Q are related via a Q-Gorenstein (qG) deformation [1].…”