Linear Systems on Irregular Varieties

Abstract

Let X be a normal complex projective variety, T ⊆ X a subvariety, a : X → A a morphism to an abelian variety such that Pic 0 (A) injects into Pic 0 (T ) and let L be a line bundle on X.Denote by X (d) → X the connectedétale cover induced by the d-th multiplication map of A, by T (d) ⊆ X (d) the preimage of T and by L (d) the pull-back of L to X (d) . For α ∈ Pic 0 (A) general, we study the restricted linear system |L (d) ⊗ a * α| |T (d) : if for some d this gives a generically finite map ϕ (d) , we show that ϕ (d) is independent of α or d sufficiently large and divisible, and is induced by the eventual map ϕ : T → Z such that a |T factorizes through ϕ.The generic value h 0 a (X |T , L) of h 0 (X |T , L ⊗ α) is called the (restricted) continuous rank. We prove that if M is the pull back of an ample divisor of A, then x → h 0 a (X |T , L + xM ) extends to a continuous function of x ∈ R, which is differentiable except possibly at countably many points; when X = T we compute the left derivative explicitly.In the case when X and T are smooth, combining the above results we prove Clifford-Severi type inequalities, i.e., geographical bounds of the form vol X|T (L) ≥ C(m)h 0 a (X |T , L), where C(m) = O(m!).

On quint‐canonical birationality of irregular threefolds

“…We now prove that . If is of maximal Albanese dimension, we know by [2, v5, Theorem 6.7] that , where general and is the least positive integer such that . Hence, .…”

“…Then we use twisted bi‐canonical or tri‐canonical system to produce a pencil of and apply Tankeev's principle, we need to show that induces a birational map on a general member to conclude that is birational. In the last step, we extensively apply a version of Reider theorem due to Langer [23] together with volume estimation of big divisors on irregular varieties, which is available due to recent progress of Severi‐type inequalities [2, 17, 19].…”

Fibered varieties over curves with low slope and sharp bounds in dimension three

“…See Remark 3.6. It is worth mentioning here that in [2,3], a very similar set of inequalities are called Clifford-Severi inequalities.…”

“…To prove (1.6), we use in an extensive way the "distinguished" section Γ 0 of f which comes from the base point of |K F |. By carefully tracing the behavior of Γ 0 under Frobenius base changes and checking the difference of canonical divisors under normalizations and resolutions of singularities, we establish a slope-like comparison between (K Xe/B + f * e A) 3 and h 0 (X e , ⌊K Xe/B ⌋) similar to (1.6), and (1.6) is a limit version of this comparison after taking e → ∞.…”

“…Notice that the fibration f : X 0 → B 0 is not necessarily relatively minimal, because it is not clear to us whether K X 0 /B 0 is still nef. 3 In the following, let…”