Let X be an arbitrary smooth irreducible complex projective curve, E→X a rank two vector bundle generated by its sections. The author first represents E as a triple {D1, D2, f},where D1, D2 are two effective divisor...Let X be an arbitrary smooth irreducible complex projective curve, E→X a rank two vector bundle generated by its sections. The author first represents E as a triple {D1, D2, f},where D1, D2 are two effective divisors with d =deg(D1) + deg(D2), and f∈H^0(X, [D1]｜D2)is a collection of polynomials. E is the extension of [D2] by [D1] which is determined by f. By using f and the Brill-Noether matrix of D1+D2, the author constructs a 2g×d matrix WE whose zero space gives Im{H^0(X, [D1])→ H^0(X, [D1]｜D1)}+Im{H^0(X, E)→H^0(X, [D2])→H^0(X, [D2]｜D2)}. From this and H^0(X, E) = H^0(X, [D1])+Im{H^0(X, E)→H^0(X, [D2])},it is got in particular that dimH^0(X, E) = deg(E)-rank(WE) + 2.展开更多
The author gives a characterization of counterexamples to the Kodaira-Ramanujam vanishing theorem on smooth projective surfaces in positive characteristic.More precisely,it is reproved that if there is a counterexampl...The author gives a characterization of counterexamples to the Kodaira-Ramanujam vanishing theorem on smooth projective surfaces in positive characteristic.More precisely,it is reproved that if there is a counterexample to the Kodaira-Ramanujam vanishing theorem on a smooth projective surface X in positive characteristic,then X is either a quasi-elliptic surface of Kodaira dimension 1 or a surface of general type.Furthermore,it is proved that up to blow-ups,X admits a fibration to a smooth projective curve,such that each fiber is a singular curve.展开更多
In this paper, we study the category H (ρ) of semi-stable coherent sheaves of a fixed slope ρ over a weighted projective curve. This category has nice properties: it is a hereditary abelian finitary length category....In this paper, we study the category H (ρ) of semi-stable coherent sheaves of a fixed slope ρ over a weighted projective curve. This category has nice properties: it is a hereditary abelian finitary length category. We will define the Ringel-Hall algebra of H (ρ) and relate it to generalized Kac-Moody Lie algebras. Finally we obtain the Kac type theorem to describe the indecomposable objects in this category, i.e. the indecomposable semi-stable sheaves.展开更多
基金Project supported by the National Natural Science Foundation of China.
文摘Let X be an arbitrary smooth irreducible complex projective curve, E→X a rank two vector bundle generated by its sections. The author first represents E as a triple {D1, D2, f},where D1, D2 are two effective divisors with d =deg(D1) + deg(D2), and f∈H^0(X, [D1]｜D2)is a collection of polynomials. E is the extension of [D2] by [D1] which is determined by f. By using f and the Brill-Noether matrix of D1+D2, the author constructs a 2g×d matrix WE whose zero space gives Im{H^0(X, [D1])→ H^0(X, [D1]｜D1)}+Im{H^0(X, E)→H^0(X, [D2])→H^0(X, [D2]｜D2)}. From this and H^0(X, E) = H^0(X, [D1])+Im{H^0(X, E)→H^0(X, [D2])},it is got in particular that dimH^0(X, E) = deg(E)-rank(WE) + 2.
基金supported by the National Natural Science Foundation of China (No. 10901037)the Doctoral Program Foundation of the Ministry of Education of China (No. 20090071120004)the Scientific Research Foundation for the Returned Overseas Chinese Scholars,State Education Ministry
文摘The author gives a characterization of counterexamples to the Kodaira-Ramanujam vanishing theorem on smooth projective surfaces in positive characteristic.More precisely,it is reproved that if there is a counterexample to the Kodaira-Ramanujam vanishing theorem on a smooth projective surface X in positive characteristic,then X is either a quasi-elliptic surface of Kodaira dimension 1 or a surface of general type.Furthermore,it is proved that up to blow-ups,X admits a fibration to a smooth projective curve,such that each fiber is a singular curve.
基金supported in part by NSF of China (Grant No. 10631010)NKBRPC (Grant No. 2006CB805905)
文摘In this paper, we study the category H (ρ) of semi-stable coherent sheaves of a fixed slope ρ over a weighted projective curve. This category has nice properties: it is a hereditary abelian finitary length category. We will define the Ringel-Hall algebra of H (ρ) and relate it to generalized Kac-Moody Lie algebras. Finally we obtain the Kac type theorem to describe the indecomposable objects in this category, i.e. the indecomposable semi-stable sheaves.
