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Let $X$ be a smooth projective variety of dimension $n$ over an algebraically closed field $k$ with ${\rm char}(k)=p>0$ and $F:X\to X_1$ be the relative Frobenius morphism. For any vector bundle $W$ on $X$, we prove that instability of... more
Let $X$ be a smooth projective variety of dimension $n$ over an algebraically closed field $k$ with ${\rm char}(k)=p>0$ and $F:X\to X_1$ be the relative Frobenius morphism. For any vector bundle $W$ on $X$, we prove that instability of $F_*W$ is bounded by instability of $W\otimes{\rm T}^{\ell}(\Omega^1_X)$ ($0\le \ell\le n(p-1)$)(Corollary \ref{cor3.8}). When $X$ is a smooth projective curve of genus $g\ge 2$, it implies $F_*W$ being stable whenever $W$ is stable.
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An open problem in algebraic geometry is the following conjecture: Let X@>f>>X ' be a complete birational morphism between two n-dimensional nondegenerate algebraic varieties. Then there exists Y such that the following... more
An open problem in algebraic geometry is the following conjecture: Let X@>f>>X ' be a complete birational morphism between two n-dimensional nondegenerate algebraic varieties. Then there exists Y such that the following diagram Y=Y↓g↓hX@>f>>X ' is commutative. O. Zariski proved in Bull. Am. Math. Soc. 48, 402-413 (1942) this conjecture for dimension n=2. Some results concerning this conjecture for dimension n=3,4 are found in papers by Moishezon, Schaps and Teicher. Some theorems for any dimension are given by V. I. Danilov [Math. USSR, Izv. 16, 419-429 (1981); translation from Izv. Akad. Nauk SSSR, Ser. Mat. 44, 465-477 (1980; Zbl 0453.14004)] and T. Luo and Z. Luo [Math. Ann. 282, No. 4, 529-534 (1988; Zbl 0672.14003)]. – Using an improved method of Teicher and Schaps the author gives the following result in dimension n=5: Theorem: Let X@>f>>X ' be a complete birational morphism between two 5- dimensional nondegenerate algebraic varieties. Le...
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Consider the surfaces of general type S with a special property: those whose canonical linear system |K S | is associated to a pencil. According to a paper of A. Beauville [Invent. Math. 55, 121-140 (1979; Zbl 0403.14006)], when the... more
Consider the surfaces of general type S with a special property: those whose canonical linear system |K S | is associated to a pencil. According to a paper of A. Beauville [Invent. Math. 55, 121-140 (1979; Zbl 0403.14006)], when the numerical invariants of the surface are large enough, the genus g of a general element of this pencil lies between 2 and 5; K 2 ≥3p g -6 when g=2, K 2 ≥(2g-2)(p g -1) otherwise. The case g=2 has been improved by the reviewer to K 2 ≥4p g -6 [“Surfaces fibrées en courbes de genre deux”, Lect. Notes Math. 1137 (1985; Zbl 0579.14028)]. The note under review gives more restrictions in the other cases, showing that K 2 is at least (24p g -56)/5, (48p g -134)/7, (80p g -262)/9, for g=3,4,5 respectively. No example is known for K 2 attaining these minimal values.
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Zariski proved around 1944 that every birational morphism between smooth surfaces over a field k is a composition of blowing-ups at closed points. Later, around 1966, Shafarevich proved the same theorem for regular schemes of dimension 2... more
Zariski proved around 1944 that every birational morphism between smooth surfaces over a field k is a composition of blowing-ups at closed points. Later, around 1966, Shafarevich proved the same theorem for regular schemes of dimension 2 without base field. This generalization is important for arithmetic geometry. Danilov generalized Zariski’s theorem to relative dimension 1 by studying the relative canonical divisor. He also left the regular scheme case as an open question. On the other hand, some results in higher dimension appeared around 1981 [see B. Crauder, Duke Math. J. 48, 589-632 (1981; Zbl 0474.14005); M. Schaps, ibid. 401-420 (1981; Zbl 0475.14008) and Math. Ann. 222, 23-28 (1976; Zbl 0309.14009); M. Teicher, ibid. 256, 391-399 (1981; Zbl 0445.14005)]. But all the authors required the algebraic varieties have an algebraically closed based field. This paper is devoted to the generalization of Schaps and Teicher’s results to regular schemes without base field. Although the ...
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Let Y be an integral Noetherian scheme. Let f:X→Y be a generically smooth, projective morphism, Cohen-Macaulay, equi-dimensional of relative dimension d and with geometrically connected fibres. Assume that X is reduced. Let ω X/Y d be the... more
Let Y be an integral Noetherian scheme. Let f:X→Y be a generically smooth, projective morphism, Cohen-Macaulay, equi-dimensional of relative dimension d and with geometrically connected fibres. Assume that X is reduced. Let ω X/Y d be the sheaf of relative regular differential forms. The aim of the paper is to study when the direct images of this sheaf are torsion-free. The question being local, it is assumed that Y=SpecD for some complete discrete valuation ring D with residue class field k algebraically closed and X=ProjS for some graded reduced D-algebra S. The authors give a set of equivalent conditions for R i f * ω X/Y d to be torsion-free. Using them the following results are proved. (1) Suppose that X/Y is arithmetically S k+2 (Serre’s condition). Then R d-i f * ω X/Y d is torsion-free and commutes with base change for all i≤k. (2) If R d-i f * ω X/Y d is torsion-free for all i≤k then R i f * O X is torsion-free for all i≤k. (3) Suppose that X/Y is globally a complete inters...
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We prove a factorization theorem of generalized functions for moduli spaces of semistable parabolic bundles of any rank.
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This article is the expanded version of a talk given at the conference: Algebraic geometry in East Asia 2008, Seoul. In this notes, I intend to give a brief survey of results on the behavior of semi-stable bundles under the Frobenius... more
This article is the expanded version of a talk given at the conference: Algebraic geometry in East Asia 2008, Seoul. In this notes, I intend to give a brief survey of results on the behavior of semi-stable bundles under the Frobenius pullback and direct images. Some results are new.
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This submission replaces the arXiv:1012.5381 submission with the same title, which had been withdrawn as it contained a mistake, repaired in this submission: on $X$ projective smooth over an algebraically closed field of characteristic... more
This submission replaces the arXiv:1012.5381 submission with the same title, which had been withdrawn as it contained a mistake, repaired in this submission: on $X$ projective smooth over an algebraically closed field of characteristic $p>0$, we show that all irreducible stratified bundles have rank 1 if and only if the commutator $[\pi_1, \pi_1]$ of the \'etale fundamental group $\pi_1$ is a pro-$p$-group, and we show that the category of stratified bundles is semi-simple with irreducible objects of rank 1 if and only if $ \pi_1 $ is abelian without $p$-power quotient. This answers positively a conjecture by Gieseker.
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Let $X$ be a smooth projective variety over an algebraically field $k$ with ${\rm char}(k)=p>0$ and $F:X\to X_1$ be the relative Frobenius morphism. When ${\rm dim}(X)=1$, we prove that $F_*W$ is a stable bundle for any stable bundle... more
Let $X$ be a smooth projective variety over an algebraically field $k$ with ${\rm char}(k)=p>0$ and $F:X\to X_1$ be the relative Frobenius morphism. When ${\rm dim}(X)=1$, we prove that $F_*W$ is a stable bundle for any stable bundle $W$ (Theorem \ref{thm1.3}). As a step to study the question for higher dimensional $X$, we generalize the canonical filtration (defined by Joshi-Ramanan-Xia-Yu for curves) to higher dimensional $X$ (Theorem \ref{thm2.6}).
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v2: A few typos corrected, a few formulations improved. On $X$ projective smooth over an algebraically closed field of characteristic $p>0$, we show that irreducible stratified bundles have rank 1 if and only if the commutator... more
v2: A few typos corrected, a few formulations improved. On $X$ projective smooth over an algebraically closed field of characteristic $p>0$, we show that irreducible stratified bundles have rank 1 if and only if the commutator $[\pi_1^{{\rm \acute{e}t}}, \pi_1^{{\rm \acute{e}t}}]$ of the \'etale fundamental group is a pro-$p$-group, and we show that the category of stratified bundles is semi-simple with
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Let $f:X\to C$ be a family of semistable K3 surfaces with non-empty set $S$ of singular fibres having infinite local monodromy. Then, when the so called Arakelov-Yau inequality reaches equality, we prove that $C\setminus S$ is a modular... more
Let $f:X\to C$ be a family of semistable K3 surfaces with non-empty set $S$ of singular fibres having infinite local monodromy. Then, when the so called Arakelov-Yau inequality reaches equality, we prove that $C\setminus S$ is a modular curve and the family comes essentially from a family of elliptic curves through a so called Nikulin-Kummer construction. In particular, when $C=\BBb
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For a stable irreducible curve $X$ and a torsion free sheaf $L$ on $X$ of rank one and degree $d$, D.S. Nagaraj and C.S. Seshadri ([NS]) defined a closed subset $\Cal U_X(r,L)$ in the moduli space of semistable torsion free sheaves of... more
For a stable irreducible curve $X$ and a torsion free sheaf $L$ on $X$ of rank one and degree $d$, D.S. Nagaraj and C.S. Seshadri ([NS]) defined a closed subset $\Cal U_X(r,L)$ in the moduli space of semistable torsion free sheaves of rank $r$ and degree $d$ on $X$. We prove that $\Cal U_X(r,L)$ is irreducible, when a smooth curve $Y$ specializes to $X$ and a line bundle $\Cal L$ on $Y$ specializes to $L$, the specialization of moduli space of semistable rank $r$ vector bundles on $Y$ with fixed determinant $\Cal L$ has underlying set $\Cal U_X(r,L)$. For rank 2 and 3, we show that there is a Cohen-Macaulay closed subscheme in the Gieseker space which represents a suitable moduli functor and has good specialization property.
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ABSTRACT The paper discusses ramification theory on arithmetic schemes. Let f:X→S be a morphism of finite type between regular schemes. Using Fitting ideals, one can define for x∈X the ramification index r(O x /O f(x) ) and the reduced... more
ABSTRACT The paper discusses ramification theory on arithmetic schemes. Let f:X→S be a morphism of finite type between regular schemes. Using Fitting ideals, one can define for x∈X the ramification index r(O x /O f(x) ) and the reduced ramification index e(O x /O f(x) ). A classical theorem of Dedekind states that r(O x /O f(x) )≥e(O x /O f(x) )-1 if X and S are one-dimensional. In the first paragraph of the paper, the above inequality is shown in the case where x and s=f(x) are points of codimension one and κ(x)/κ(s) is separably generated. For suitable arithmetic S-schemes X, this inequality gives information on the ramification divisor of X/S by relating the latter to the relative canonical sheaf κ X/S . In the second paragraph, S is the spectrum of a complete discrete valuation ring with algebraically closed residue class field and X is a proper regular arithmetic surface over S. The Artin conductor of X/S is by definition Art(X/S)=χ(X s )-χ(X η ¯ )-sw(X/S), where χ(X s ) and χ(X η ¯ ) are the Euler characteristics of the special and the geometric generic fibre of X and sw(X/S) is the Swan conductor. S. Bloch [in Algebraic geometry, Brunswick/Maine 1985, Part 2, Proc. Sympos. Pure Math. 46, 421–450 (1987; Zbl 0654.14004)] has expressed Art(X/S) as the degree of a suitable localized Chern class. The author uses this description to show that Art(X/S)=r(X/S), where r(X/S) is the ramification number of X/S which is defined purely in terms of the Fitting ideal sheaf ℱ X/S and the canonical sheaf κ X/S . The last paragraph describes for arithmetic surfaces X/S the change of the numerical invariants κ X/S 2 and degf * κ X/S under base change. This is used to re-prove a height inequality of S.-L. Tan [J. Reine Angew. Math. 461, 123–135 (1995; Zbl 0814.14028)].
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For any smooth projective variety X of dimension n over an algebraically closed field k of characteristic p>0 with μ(ΩX1)>0, if the truncated symmetric powers Tℓ(ΩX1) (0ℓn(p−1)) of ΩX1 are semi-stable, then the sheaf BX1 of exact... more
For any smooth projective variety X of dimension n over an algebraically closed field k of characteristic p>0 with μ(ΩX1)>0, if the truncated symmetric powers Tℓ(ΩX1) (0ℓn(p−1)) of ΩX1 are semi-stable, then the sheaf BX1 of exact 1-forms is stable. When X is a surface with μ(ΩX1)>0 and ΩX1 is semi-stable, the sheaf BX2 of exact 2-forms is also stable.