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UNIFORMITY OF HOLOMORPHIC VECTOR BUNDLES ON INFINITE-DIMENSIONAL FLAG MANIFOLDS
Ballico, E. Korean Mathematical Society 2003 대한수학회보 Vol.40 No.1
Let V be a localizing infinite-dimensional complex Banach space. Let X be a flag manifold of finite flags either of finite codimensional closed linear subspaces of V or of finite dimensional linear subspaces of V. Let E be a holomorphic vector bundle on X with finite rank. Here we prove that E is uniform, i.e. that for any two lines $D_1$ R in the same system of lines on X the vector bundles E$\mid$D and E$\mid$R have the same splitting type.
BRILL-NOETHER THEORY FOR RANK 1 TORSION FREE SHEAVES ON SINGULAR PROJECTIVE CURVES
Ballico, E. Korean Mathematical Society 2000 대한수학회지 Vol.37 No.3
Let X be an integral Gorenstein projective curve with g:=pa(X) $\geq$ 3. Call $G^r_d$ (X,**) the set of all pairs (L,V) with L$\epsilon$Pic(X), deg(L) = d, V $\subseteq$ H^0$(X,L), dim(V) =r+1 and V spanning L. Assume the existence of integers d, r with 1 $\leq$ r$\leq$ d $\leq$ g-1 such that there exists an irreducible component, , of $G^r_d$(X,**) with dim($\Gamma$) $\geq$ d - 2r and such that the general L$\geq$$\Gamma$ is spanned at every point of Sing(X). Here we prove that dim( ) = d-2r and X is hyperelliptic.
Holomorphic embeddings of Stein spaces in infinite-dimensional projective spaces
E. Ballico 대한수학회 2005 대한수학회지 Vol.42 No.1
Let X be a reduced Stein space and L a holomorphic line bundle on X. L is spanned by its global sections and the associated holomorphic map hL : X ! P(H0(X;L)¤) is an embed-ding. Choose any locally convex vector topology ¿ on H0(X;L)¤ stronger than the weak-topology. Here we prove that hL(X) is sequentially closed in P(H0(X;L)¤) and arithmetically Cohen - Macaulay, i.e. for all integers k ¸ 1 the restriction map ½k : H0(P(H0(X;L)¤);OP(H0(X;L)¤)(k)) ! H0(hL(X);OhL(X)(k)) »= H0(X;Lk) is surjective.
HOLOMORPHIC EMBEDDINGS OF STEIN SPACES IN INFINITE-DIMENSIONAL PROJECTIVE SPACES
BALLICO E. Korean Mathematical Society 2005 대한수학회지 Vol.42 No.1
Lpt X be a reduced Stein space and L a holomorphic line bundle on X. L is spanned by its global sections and the associated holomorphic map $h_L\;:\;X{\to}P(H^0(X, L)^{\ast})$ is an embedding. Choose any locally convex vector topology ${\tau}\;on\;H^0(X, L)^{\ast}$ stronger than the weak-topology. Here we prove that $h_L(X)$ is sequentially closed in $P(H^0(X, L)^{\ast})$ and arithmetically Cohen -Macaulay. i.e. for all integers $k{\ge}1$ the restriction map ${\rho}_k\;:\;H^0(P(H^0(X, L)^{\ast}),\;O_{P(H^0(X, L)^{\ast})}(k)){\to}H^0(h_L(X),O_{hL_(X)}(k)){\cong}H^0(X, L^{\otimes{k}})$ is surjective.
Uniformity of holomorphic vector bundles on infinite-dimensional flag manifolds
E. Ballico 대한수학회 2003 대한수학회보 Vol.40 No.1
Let V be a localizing infinite-dimensional complex Banach space. Let X be a flag manifold of finite flags either offinite codimensionalclosed linear subspaces of V or of finite dimensional linear subspaces of V. Let E be a holomorphicvector bundle on X with finite rank. Here we prove that E is uniform, i.e. that for any two lines D, R in the samesystem of lines on X the vector bundles Evert D and Evert R have the same splitting type.
Globally generated vector bundles of rank 2 on a smooth quadric threefold
Ballico, E.,Huh, S.,Malaspina, F. North-Holland Pub. Co 2014 Journal of pure and applied algebra Vol.218 No.2
We investigate the existence of globally generated vector bundles of rank 2 with c<SUB>1</SUB>@?3 on a smooth quadric threefold and determine their Chern classes. As an automatic consequence, every rank 2 globally generated vector bundle on Q with c<SUB>1</SUB>=3 is an odd instanton up to twist.
Ballico, E.,Huh, S.,Malaspina, F. Academic Press 2016 Journal of algebra Vol.450 No.-
<P>We classify globally generated vector bundles on P-1 x P-1 x P-1 with small first Chern class, i.e. c(1) = (a(1), a(2), a(3)), a(i) <= 2. Our main method is to investigate the associated smooth curves to globally generated vector bundles via the Hartshorne-Serre correspondence. (C) 2015 Elsevier Inc. All rights reserved.</P>