Algebraic surfaces of general type with $c^2_1=3p_g-7$

Certain algebraic surfaces of general type with irregularity one and their canonical mappings

“…In §4.2.2, we proved the existence of a surface S with K 2 S = 3p g (S), q(S) = 1 and p g (S) = 2, but did not study the canonical mapping Φ |K S | in the case E ∼ = E 0 ⊕L, (E 0 ∈ E C (2, 1), L ∈ E C (1, 1)). On the other hand, we showed the existence of a surface S with the same invariants in the case E ∈ E C (3,2). We obtain the following result in these two cases: Proposition 4.41 Let E be one of the following:…”

“…Since the complete linear system of O Y (4C 0 ) ⊗ µ * (det E ∨ 0 ) ⊗2 is a pencil without base points, any base point of |4C 0 − µ * D| exists only on Γ : 3 are pairwise different, and since F 1 , F 2 and F 3 intersect Γ at different points, we obtain…”

On surfaces with a canonical pencil

“…1 We consider the sheaf of algebras S := A/x 2 1 . Since S = A/x 1 , the surjection A → S correspond to the inclusion of the non reduced divisor 2s ⊂ C.…”

“…Example 4. 9 We give examples for each pair (α, ) with 1 ≤ α ≤ ≤ 2, p g = 2α − +4: three examples with (α, , K 2 , p g ) which equals respectively (1,1,15,5), (1,2,12,4) and (2,2,20,6).…”

Even canonical surfaces with small K2, I

“…In other words, canonical surfaces with K < Ap g -12 should have a flavor similar to the exceptions of Enriques-Babbage-Petri's theorem, i.e., trigonal curves and plane quintic curves. Unfortunately, as of now, the conjecture is known to hold only for canonical surfaces with K = 3p g -7, 3p g -6 (see [4], [8], [1], [10] and [12]). …”