Variations on the Weak Bounded Negativity Conjecture

Theorem 1

Let Y be the blow-up of ℙ² at n arbitrary (proper or infinitely near) points and let L be the pull-back of a (general) line in ℙ². Then all but finitely many reduced and irreducible curves C on Y satisfy the inequality

Theorem 2

Let Y be the blow-up of ℙ² at n arbitrary (proper or infinitely near) points and let L be the pull-back of a (general) line in ℙ². Then all but finitely many reduced and irreducible curves C on Y satisfy the inequality

Conjecture 3

(Vojta). Let X be a smooth projective surface. There exist constants A, B such that for any integral curve C we have k_C ≤ A(g − 1) + B.

Theorem 4

Let X be a smooth projective surface with h⁰(X, −K_X) = 0 and let C be a smooth irreducible curve of genus g on X.

1. If h⁰(X, 2K_X + C) ≠ 0 then k_C ≤ 4(g − 1) + 3 c₂(X) − KX2 .

2. If h⁰(X, 2K_X + C) = 0 then k_C ≤ − (g − 1) − KX2 − χ (𝓞_X).

Theorem 5

Let X be a smooth projective surface with h⁰(X, −K_X) = 0 and let C be an integral curve of geometric genus g on X. If h⁰(X, 2K_X + C) = 0 then

Lemma 6

Let X be a smooth projective surface and let C be a reduced and irreducible curve on X. Then for every integer m ≠ 1 we have

Proof

Just apply the Riemann–Roch theorem to mK_X + C, Serre duality to h²(mK_X + C) and the adjunction formula to C:

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Lemma 7

Notation as in Lemma 6. If h⁰(−mK_X) ≠ 0 for some m ≥ 1, then all but finitely many reduced and irreducible curves C on X satisfy the inequality C² ≥ − 2.

Proof

Let E be an effective divisor in |−mK_X|. If C is not one of the finitely many curves in the support of E then −K_X.C ≥ 0, hence C² = −K_X.C + 2p_a(C) − 2 ≥ − 2. □

Lemma 8

Notation as in Theorem 1. Let m₀ be the integer such that

Then h⁰(m₀ K_Y + C) = 0.

Proof

We have (m₀ K_Y + C).L = − 3 m₀ + C.L ≤ − 1 < 0 and L is nef, hence the divisor m₀ K_Y + C cannot be effective. □

Proof of Theorem 1

We apply Lemma 6 to X = Y, in particular we have χ(𝓞_Y) = 1 and KY2 = 9 − n. By Lemma 7 we may assume that h⁰(−(m − 1)K_Y − C) = h⁰(−(m − 1)K_Y) = 0. By setting m = m₀ as in Lemma 8 we obtain

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Proof of Theorem 2

If h⁰(2 K_Y + C) = 0, then by arguing as in the proof of Theorem 1 with m = 2 we obtain C² ≥ min {−n, −2}. Assume now h⁰(2 K_Y + C) ≠ 0, so that in particular h⁰(2 (K_Y + C)) ≠ 0. By Lemma 7 we may also assume h⁰(− K_Y) = 0. Hence we are in the position to apply [5], Corollary 1.8, and deduce

On the other hand, by arguing as in the proof of Theorem 1, we obtain

It follows that

and we conclude

□

Proof of Theorem 5

The idea is to blow up X resolving step by step the singularities of C. If the assumptions hold at each step provided they hold at the previous one and the conclusion holds at each step provided it holds at the next one, then the statement follows recursively from the smooth case, namely from item (b) of Theorem 4.

Let X be a smooth projective surface and let C be an integral curve of geometric genus g on X. Let p ∈ C be a point with multiplicity mult_p(C) = m ≥ 2 and let π: X͂ → X be the blow-up of X at p. Let E be the exceptional divisor of the blow-up and let C͂ = π^*(C) − mE be the strict transform of C.

We claim the following:

1. If h⁰(X, −K_X) = 0 then h⁰(X͂, −K_X͂) = 0.

2. If h⁰(X, 2K_X + C) = 0 then h⁰(X͂, 2K_X͂ + C͂) = 0.

3. If k_C͂ ≤ − (g − 1) − KX~2 − χ (𝓞_X͂) then k_C ≤ − (g − 1) − KX2 − χ (𝓞_X).

Indeed, (i) follows from K_X͂ = π^*(K_X) + E (for details see [5], Lemma 1.10).

Next, for (ii) we have h⁰(X͂, 2K_X͂ + C͂) = h⁰(X͂, 2 (π^*(K_X) + E) + π^*(C) − mE)) ≤ h⁰(X͂, 2 π^*(K_X) + π^*(C)) = h⁰(X͂, π^*(2K_X + C)) = h⁰(X, 2K_X + C) = 0.

Finally, we have k_C = K_X . C = K_X͂ . C͂ − m = k_C͂ − m ≤ − (g − 1) − KX~2 − χ (𝓞_X͂) − m = − (g − 1) − ( KX2 − 1) − χ (𝓞_X) − m < −(g − 1) − KX2 − χ (𝓞_X). □

Remark 9

If the curve C = ∑in C_i is reducible (but still reduced), the argument above works verbatim by setting g : = ∑in g_i − (n − 1), where g_i is the geometric genus of the irreducible component C_i. Indeed, after finitely many blow-ups we obtain a curve C͂ which is the disjoint union of the normalizations of the curves C_i. The arithmetic genus of C͂ is p_a(C͂) = ∑in g_i − (n − 1) = g and the proof of [5], Lemma 1.3, implies C͂² ≥ 3 p_a(C͂) + KX~2 + χ (𝓞_X͂) − 3, hence we have K_X͂ . C͂ ≤ − (p_a(C͂) − 1) − KX~2 − χ (𝓞_X͂) = − (g − 1) − KX~2 − χ (𝓞_X͂), exactly as in the integral case.

Remark 10

Under the same assumptions, the argument above yields the following stronger inequality:

Remark 11

The argument above does not apply to the extension to singular curves of item (a) of Theorem 4 because the analogue of item (iii) does not work. Indeed, if k_C͂ ≤ 4(g − 1) + 3 c₂(X͂) − KX~2 then k_C = k_C͂ − m ≤ 4(g − 1) + 3(c₂(X) + 1) − ( KX2 − 1) − m = 4(g − 1) + 3 c₂(X) − KX2 + (4 − m), with 4 − m > 0 if 2 ≤ m ≤ 3.