On algebraic surfaces of general type with negative

Abstract

Abstract. We prove that for any prime number p ≥ 3, there exists a positive number κp such that χ(O X ) ≥ κpc 2 1 holds true for all algebraic surfaces X of general type in characteristic p. In particular, χ(O X ) > 0. This answers a question of N. Shepherd-Barron when p ≥ 3.

Slope inequalities and a Miyaoka-Yau type inequality

“…To end this section, it is worthing to mention that Theorem 3.2 implies Gu's conjecture for the "hyperelliptic part" (see Conjecture 1.4 of [6]).…”

“…Shepherd-Barron also suggested that the most obvious place to look for such examples would be in the case where (p, g) = (2, 2). Later, it is proved by the first author in [6] that χ(O S ) > 0 when p ≥ 3. Our main result in this article is to prove a Miyaoka-Yau type inequality:…”

“…Note that π : P → C is birational to a ruled surface (recall that K(C) is C1 by Tsen's Theorem) and P ′ is normal with integral π-fibres, we see that P ′ is exactly a smooth minimal ruled surface over C. Thus the quotient map σ : S → P ′ is a flat double cover with some branch divisor Σ ′ P ′ , and Σ ′ itself is smooth over k (see [6, § 2.2]). on the other hand, it can be deduced from [6…”

Integral forms and torsors of inseparable forms of G_a

“…If g > 1, the surface X is of general type and a recent work of Yi Gu shows that χ(X, O X ) > 0 [10], [11] (eliminating a few possible cases where it may be not true from [31]). Thus in this case, deg R 1 f * O X < 0.…”

Toward good families of codes from towers of surfaces

“…Remark 3.2. Note that K Xn being ample, we have κ 0, and that χ 0 in case q is odd, since from [25] we have χ(O X ) > 0 for any surface X of general type in odd characteristic.…”