C. Keem - Academia.edu (original) (raw)
Papers by C. Keem
Results in Mathematics, 2004
ABSTRACT Let f: X → C be a “ general ” double covering of a smooth curve C of genus h ≥ 1. Here w... more ABSTRACT Let f: X → C be a “ general ” double covering of a smooth curve C of genus h ≥ 1. Here we show (with some restrictions on C if h ≥ 2) that there is no P ∈ X which is a common ramification point of all degree calG−2h+1{cal G} - 2h + 1calG−2h+1 morphisms X → P1.
Mathematische Annalen, 1990
The Jacobian variety J(C) of a smooth complex algebraic curve C of genus g is a g-dimensional abe... more The Jacobian variety J(C) of a smooth complex algebraic curve C of genus g is a g-dimensional abelian variety which parametrizes all the line bundles of given degree d on C. We denote by Wj(C) the locus in J(C) corresponding to those line bundles of degree d with r + 1 or more independent global sections. Then Wd'(C) is an analytic subvariety of J(C) and can equivalently be viewed as the subvariety consisting of all effective divisor classes of degree d which move in a linear system of projective dimension at least r. Ifd > g + r-2, one can compute the dimension of WJ(C) by using the Riemann-Roch formula, and this dimension is independent of C. If d<g+r-2, the dimension of War(C) is known to be equal to the Brill-Noether number ~(d, g, r): = g-(r+l)(g-d+r) for a general curve C by a theorem of Griffiths and Harris [GH 1], but dim W~'(C) might be greater than Q(d, g, r) for some special curve C. So one can ask for the maximum dimension of W~(C) for d < g + r-2. The answer to this was provided by Martens in [M 1]: (0.1) Proposition (Martens). Let C be a smooth algebraic curve of genus g > 3. Let d and r be integers such that d < g + r-2, r > 1. Then dim WJ(C) < d-2r and equality holds if and only if C is hyperelliptic. One can then ask for a description of those non-hyperelliptic curves C which achieve the maximum value d-2r-1 of the dimension of 14~a(C). This was answered by Mumford in [Mu 1]: * During the period when this manuscript was prepared for publication, the author was visiting the Max-Planck-Institut fiir Mathematik in Bonn, FRG to which he is grateful
Results in Mathematics, 2004
ABSTRACT Let f: X → C be a “ general ” double covering of a smooth curve C of genus h ≥ 1. Here w... more ABSTRACT Let f: X → C be a “ general ” double covering of a smooth curve C of genus h ≥ 1. Here we show (with some restrictions on C if h ≥ 2) that there is no P ∈ X which is a common ramification point of all degree calG−2h+1{cal G} - 2h + 1calG−2h+1 morphisms X → P1.
Journal für die reine und angewandte Mathematik (Crelles Journal), 1990
Japan 251 Japanese Empire, colonies, dependencies, etc. (Collectively) 253 North Japan.
Mathematische Zeitschrift, 1998
Manuscripta Mathematica, 1992
In this paper we study a new numerical invariant ~ of curves C whidl is related to the primitive ... more In this paper we study a new numerical invariant ~ of curves C whidl is related to the primitive linear series on C. (Primitive series-defined below-are the essential complete and special linear series on C.) The curves with/~ _< 3 are classified, and it is shown that for a given value of s the curve is a double covering if its genus is sufficiently high. The main tool are dilnension theorems of It. Martens-Mumfordtype for the varieties of special divisors of C, and we prove two refinements of these theorems.
Manuscripta Mathematica, 1989
In this paper ~ve give an alternative proof of the fact that a general (e+2)-gonal curve of genus... more In this paper ~ve give an alternative proof of the fact that a general (e+2)-gonal curve of genus g_>2e+2 has Clifford index e. This was conjectured by M. Green and I%. Lazarsfeld in [G-L] and ~vas later proved by Ballico in []5] using the technique of the limit linear series. Here %re prove a lemma %vhich gives an upper bound on the dimension of the variety of special linear systems on a variable curve and then proceed to prove the theorem of Ballico using this lemma.
Kodai Mathematical Journal, 2005
ABSTRACT We show the existence of special curves which are ramified coverings of irrational curve... more ABSTRACT We show the existence of special curves which are ramified coverings of irrational curves with surjective Wahl maps. We also show the failure of a key property of Gaussian maps in positive characteristic, i.e. the Gherardelli's lower bound for the rank of a Gaussian map.
Annali di Matematica Pura ed Applicata, 1998
In this paper we study finite sets of smooth algebraic curves which are the support of special di... more In this paper we study finite sets of smooth algebraic curves which are the support of special divisors (,~Weierstrass sets,). We prove several existence results of Weierstrass sets with low weight on suitable curves (e.g. general k-gonal curves). Recently (see [K1], [Ho], [Is] and [BKi]) several papers studied the natural generalization of the notion of Weierstrass point of a smooth projective curve C, considering instead of the ,exceptional points, of C the ,,exceptional finite subsets-or ,Weierstrass subsets, of C. The following very natural definition was introduced in [Bt~].
Here we prove the projective normality of several special line bundles on a general k-gonal curve... more Here we prove the projective normality of several special line bundles on a general k-gonal curve. Let X be a k-gonal curve arising as the normalization of a certain nodal curve Y ⊂ P 1 × P 1. We prove the existence of many projectively normal special line bundles on X. We also show the existence of a large set, Φ, of special line bundles on X which are not projectively normal (and not even quadratically normal) and for every L ∈ Φ we compute the dimension of the cokernel of the multiplication map H 0 (X, L) ⊗ H 0 (X, L) → H 0 (X, L ⊗2). Let M be the blowing-up either of P 2 or of P 1 × P 1 at a general finite set S. We show the projective normality of certain line bundles on M , the case P 1 ×P 1 being used to prove our results on k-gonal curves. 1. Introduction. Let X be a smooth k-gonal curve of genus g and R ∈ Pic k (X) its degree k pencil. We assume h 0 (X, R ⊗t) = t + 1 if 0 ≤ t ≤ [g/(k − 1)]
International Mathematical Forum, 2006
For all integers r ≥ 2 and any smooth and connected projective curve X, let ρ X (r) denote the mi... more For all integers r ≥ 2 and any smooth and connected projective curve X, let ρ X (r) denote the minimal integer d such that there is a morphism φ : X → P r birational onto its image and such that deg(φ(X)) = d and φ(X) spans P r. Fix integers d, g such that d ≥ 8 and d 2 /6 < g ≤ d 2 /4−d. Here we prove the existence of a smooth genus g curve X such that ρ X (3) = d.
Journal of Pure and Applied Algebra, 2006
In [KKM] it was seen that a linear series greC2r computing the Clifford index e of an algebraic c... more In [KKM] it was seen that a linear series greC2r computing the Clifford index e of an algebraic curve C is birationally very ample if r 3 and e 3. The purpose of this present note is to make further observations along the same lines. We also introduce the notion of higher order Clifford indices and
computing the Cliffordindex e of an algebraic curve C is birationally very ample if r 3ande3.Th... more computing the Cliffordindex e of an algebraic curve C is birationally very ample if r 3ande3.The purpose of this present note is to make further observations along thesame lines. We also introduce the notion of higher order Clifford indices andmake a few remarks on it.We first fix some basic terminology and notations. C always denote asmooth irreducible projective curve of genus g 4. A g
Fix integers q, g, k, d. Set πd,k,q := kd − d − k + kq + 1 and assume q > 0, k ≥ 2, d ≥ 3q + 1... more Fix integers q, g, k, d. Set πd,k,q := kd − d − k + kq + 1 and assume q > 0, k ≥ 2, d ≥ 3q + 1, g ≥ kq − k + 1 and πd,k,q − ((⌊d/2⌋ + 1 − q) · (⌊k/2⌋ + 1) ≤ g ≤ πd,k,q. Let Y be a smooth and connected genus q projective curve. Here we prove the existence of a smooth and connected genus g projective curve X, a degree k morphism f : X → Y and a degree d morphism u : X → P such that the morphism (f, u) : X → Y × P is birational onto its image.
Proceedings of the American Mathematical Society
Let X be a general k-gonal curve of genus g .H ere we prove a strong upper bound for the dimensio... more Let X be a general k-gonal curve of genus g .H ere we prove a strong upper bound for the dimension of linear series on X, i.e. we prove that dim(W r
Journal of Pure and Applied Algebra, 2006
We prove various properties of varieties of special linear systems on double coverings of hyperel... more We prove various properties of varieties of special linear systems on double coverings of hyperelliptic curves. We show and determine the irreducibility, generically reducedness and singular loci of the variety W r d for bi-elliptic curves and double coverings of genus two curves. Similar results for double coverings of hyperelliptic curves of genus h ≥ 3 are also presented.
Results in Mathematics, 2004
ABSTRACT Let f: X → C be a “ general ” double covering of a smooth curve C of genus h ≥ 1. Here w... more ABSTRACT Let f: X → C be a “ general ” double covering of a smooth curve C of genus h ≥ 1. Here we show (with some restrictions on C if h ≥ 2) that there is no P ∈ X which is a common ramification point of all degree calG−2h+1{cal G} - 2h + 1calG−2h+1 morphisms X → P1.
Mathematische Annalen, 1990
The Jacobian variety J(C) of a smooth complex algebraic curve C of genus g is a g-dimensional abe... more The Jacobian variety J(C) of a smooth complex algebraic curve C of genus g is a g-dimensional abelian variety which parametrizes all the line bundles of given degree d on C. We denote by Wj(C) the locus in J(C) corresponding to those line bundles of degree d with r + 1 or more independent global sections. Then Wd'(C) is an analytic subvariety of J(C) and can equivalently be viewed as the subvariety consisting of all effective divisor classes of degree d which move in a linear system of projective dimension at least r. Ifd > g + r-2, one can compute the dimension of WJ(C) by using the Riemann-Roch formula, and this dimension is independent of C. If d<g+r-2, the dimension of War(C) is known to be equal to the Brill-Noether number ~(d, g, r): = g-(r+l)(g-d+r) for a general curve C by a theorem of Griffiths and Harris [GH 1], but dim W~'(C) might be greater than Q(d, g, r) for some special curve C. So one can ask for the maximum dimension of W~(C) for d < g + r-2. The answer to this was provided by Martens in [M 1]: (0.1) Proposition (Martens). Let C be a smooth algebraic curve of genus g > 3. Let d and r be integers such that d < g + r-2, r > 1. Then dim WJ(C) < d-2r and equality holds if and only if C is hyperelliptic. One can then ask for a description of those non-hyperelliptic curves C which achieve the maximum value d-2r-1 of the dimension of 14~a(C). This was answered by Mumford in [Mu 1]: * During the period when this manuscript was prepared for publication, the author was visiting the Max-Planck-Institut fiir Mathematik in Bonn, FRG to which he is grateful
Results in Mathematics, 2004
ABSTRACT Let f: X → C be a “ general ” double covering of a smooth curve C of genus h ≥ 1. Here w... more ABSTRACT Let f: X → C be a “ general ” double covering of a smooth curve C of genus h ≥ 1. Here we show (with some restrictions on C if h ≥ 2) that there is no P ∈ X which is a common ramification point of all degree calG−2h+1{cal G} - 2h + 1calG−2h+1 morphisms X → P1.
Journal für die reine und angewandte Mathematik (Crelles Journal), 1990
Japan 251 Japanese Empire, colonies, dependencies, etc. (Collectively) 253 North Japan.
Mathematische Zeitschrift, 1998
Manuscripta Mathematica, 1992
In this paper we study a new numerical invariant ~ of curves C whidl is related to the primitive ... more In this paper we study a new numerical invariant ~ of curves C whidl is related to the primitive linear series on C. (Primitive series-defined below-are the essential complete and special linear series on C.) The curves with/~ _< 3 are classified, and it is shown that for a given value of s the curve is a double covering if its genus is sufficiently high. The main tool are dilnension theorems of It. Martens-Mumfordtype for the varieties of special divisors of C, and we prove two refinements of these theorems.
Manuscripta Mathematica, 1989
In this paper ~ve give an alternative proof of the fact that a general (e+2)-gonal curve of genus... more In this paper ~ve give an alternative proof of the fact that a general (e+2)-gonal curve of genus g_>2e+2 has Clifford index e. This was conjectured by M. Green and I%. Lazarsfeld in [G-L] and ~vas later proved by Ballico in []5] using the technique of the limit linear series. Here %re prove a lemma %vhich gives an upper bound on the dimension of the variety of special linear systems on a variable curve and then proceed to prove the theorem of Ballico using this lemma.
Kodai Mathematical Journal, 2005
ABSTRACT We show the existence of special curves which are ramified coverings of irrational curve... more ABSTRACT We show the existence of special curves which are ramified coverings of irrational curves with surjective Wahl maps. We also show the failure of a key property of Gaussian maps in positive characteristic, i.e. the Gherardelli's lower bound for the rank of a Gaussian map.
Annali di Matematica Pura ed Applicata, 1998
In this paper we study finite sets of smooth algebraic curves which are the support of special di... more In this paper we study finite sets of smooth algebraic curves which are the support of special divisors (,~Weierstrass sets,). We prove several existence results of Weierstrass sets with low weight on suitable curves (e.g. general k-gonal curves). Recently (see [K1], [Ho], [Is] and [BKi]) several papers studied the natural generalization of the notion of Weierstrass point of a smooth projective curve C, considering instead of the ,exceptional points, of C the ,,exceptional finite subsets-or ,Weierstrass subsets, of C. The following very natural definition was introduced in [Bt~].
Here we prove the projective normality of several special line bundles on a general k-gonal curve... more Here we prove the projective normality of several special line bundles on a general k-gonal curve. Let X be a k-gonal curve arising as the normalization of a certain nodal curve Y ⊂ P 1 × P 1. We prove the existence of many projectively normal special line bundles on X. We also show the existence of a large set, Φ, of special line bundles on X which are not projectively normal (and not even quadratically normal) and for every L ∈ Φ we compute the dimension of the cokernel of the multiplication map H 0 (X, L) ⊗ H 0 (X, L) → H 0 (X, L ⊗2). Let M be the blowing-up either of P 2 or of P 1 × P 1 at a general finite set S. We show the projective normality of certain line bundles on M , the case P 1 ×P 1 being used to prove our results on k-gonal curves. 1. Introduction. Let X be a smooth k-gonal curve of genus g and R ∈ Pic k (X) its degree k pencil. We assume h 0 (X, R ⊗t) = t + 1 if 0 ≤ t ≤ [g/(k − 1)]
International Mathematical Forum, 2006
For all integers r ≥ 2 and any smooth and connected projective curve X, let ρ X (r) denote the mi... more For all integers r ≥ 2 and any smooth and connected projective curve X, let ρ X (r) denote the minimal integer d such that there is a morphism φ : X → P r birational onto its image and such that deg(φ(X)) = d and φ(X) spans P r. Fix integers d, g such that d ≥ 8 and d 2 /6 < g ≤ d 2 /4−d. Here we prove the existence of a smooth genus g curve X such that ρ X (3) = d.
Journal of Pure and Applied Algebra, 2006
In [KKM] it was seen that a linear series greC2r computing the Clifford index e of an algebraic c... more In [KKM] it was seen that a linear series greC2r computing the Clifford index e of an algebraic curve C is birationally very ample if r 3 and e 3. The purpose of this present note is to make further observations along the same lines. We also introduce the notion of higher order Clifford indices and
computing the Cliffordindex e of an algebraic curve C is birationally very ample if r 3ande3.Th... more computing the Cliffordindex e of an algebraic curve C is birationally very ample if r 3ande3.The purpose of this present note is to make further observations along thesame lines. We also introduce the notion of higher order Clifford indices andmake a few remarks on it.We first fix some basic terminology and notations. C always denote asmooth irreducible projective curve of genus g 4. A g
Fix integers q, g, k, d. Set πd,k,q := kd − d − k + kq + 1 and assume q > 0, k ≥ 2, d ≥ 3q + 1... more Fix integers q, g, k, d. Set πd,k,q := kd − d − k + kq + 1 and assume q > 0, k ≥ 2, d ≥ 3q + 1, g ≥ kq − k + 1 and πd,k,q − ((⌊d/2⌋ + 1 − q) · (⌊k/2⌋ + 1) ≤ g ≤ πd,k,q. Let Y be a smooth and connected genus q projective curve. Here we prove the existence of a smooth and connected genus g projective curve X, a degree k morphism f : X → Y and a degree d morphism u : X → P such that the morphism (f, u) : X → Y × P is birational onto its image.
Proceedings of the American Mathematical Society
Let X be a general k-gonal curve of genus g .H ere we prove a strong upper bound for the dimensio... more Let X be a general k-gonal curve of genus g .H ere we prove a strong upper bound for the dimension of linear series on X, i.e. we prove that dim(W r
Journal of Pure and Applied Algebra, 2006
We prove various properties of varieties of special linear systems on double coverings of hyperel... more We prove various properties of varieties of special linear systems on double coverings of hyperelliptic curves. We show and determine the irreducibility, generically reducedness and singular loci of the variety W r d for bi-elliptic curves and double coverings of genus two curves. Similar results for double coverings of hyperelliptic curves of genus h ≥ 3 are also presented.