On Singular Cubic Surfaces

Abstract

Abstract. We study global log canonical thresholds of singular cubic surfaces.Key words. Cubic surfaces, singularities, log canonical thresholds, del Pezzo fibrations, birational maps, Kahler-Einstein metric, alpha-invariant of Tian, orbifolds.AMS subject classifications. 14J26, 14J45, 14J70, 14Q10, 14B05, 14E05, 32Q20All varieties are assumed to be defined over C.1. Introduction. Let X be a variety with at most log terminal singularities, let Z ⊆ X be a closed subvariety, and let D be an effective Q-Cartier Q-divisor on X. Then the number lct Z X, D = sup λ ∈ Q the log pair X, λD is log canonical along Z is said to be the log canonical threshold of D along Z (see [8]).For the case Z = X we use the notation lct(X, D) instead of lct X (X, D). Then lct X, D = inf lct P X, D P ∈ X = sup λ ∈ Q the log pair X, λD is log canonical .Suppose, in addition, that X is a Fano variety. Definition 1.2. We define the global log canonical threshold of X by the numberThe number lct(X) is an algebraic counterpart of the α-invariant introduced in [11]. In this paper we prove the following result 1 .Theorem 1.4. Let X be a singular cubic surface in P 3 with canonical singularities. Then such that π andπ are flat morphisms, and ρ is a birational map that induces an isomorphismwhere X andX are scheme fibers of π andπ over a point O ∈ Z, respectively. Suppose that • the varieties V andV have terminal Q-factorial singularities,• the divisors −K V and −KV are π-ample andπ-ample, respectively, • the fibers X andX are irreducible. Then ρ is an isomorphism if one of the following conditions hold:• the varieties X andX have log terminal singularities, and lct(X)+lct(X) > 1;• the variety X has log terminal singularities, and lct(X) 1.The assertion of Theorem 1.5 is sharp (see [10, Example 5.2-5.6]).Example 1.8. Let V beV subvarieties in C 1 × P 3 given by the equations x 3 + y 3 + z 2 w + t 6 w 3 = 0 and x 3 + y 3 + z 2 w + w 3 = 0, respectively, where t is a coordinate on C 1 , and (x, y, z, w) are coordinates on P 3 . The projections π : V −→ C 1 andπ :V −→ C 1 1 A cubic surface in P 3 with isolated singularities has canonical singularities ⇐⇒ it is not a cone.

Birational geometry of del Pezzo fibrations with terminal quotient singularities

“…We generalize [, Theorem 1.5] for our purposes. Theorem Let be a ‐Fano fibration over a curve .…”

Deformations of nodal Kähler-Einstein Del Pezzo surfaces with discrete automorphism groups

“…Moreover, it is known that that C 0 admits an orbifold KE metric (e.g., [8], or simply by noting that it is a quotient of the KE Del Pezzo surface of degree 6). Thus, considering generic deformations of C 0 of the form…”

Log canonical thresholds on Gorenstein canonical del Pezzo surfaces

“…Therefore, the curve C consists of only lines. We also see that the first form must be (2, 2, 2) and the second must be (3,2). For the case (2, 2, 2), the curve C must consists of one double line and one single line.…”

“…For the case (3,2), the curve C must also consists of one double line and one single line. We must take the blow up at the intersection point of the double line and the single line.…”