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1.
Robin Hartshorne 《Annali dell'Universita di Ferrara》1994,40(1):207-223
LetC be a curve contained in ℙ
k
3
(k of any characteristic), which is locally Cohen-Macaulay, not contained in a plane and of degreed. We prove thatp
a
(C)≤≤((d−2)(d−3))/2. Moreover we show existence of curves with anyd, p
a
satisfying this inequality and we characterize those curves for which equality holds.
Sunto Si dimostra che seC⊃ℙ k 3 (k di caratteristica qualunque) é una curva, non piana, localmente Cohen-Macaulay, di gradod, allorap a (C)≤((d−2)(d−3))/2. Si mostra che questa limitazione è ottimale, e si classificano le curve di genere aritmetico massimale.相似文献
2.
Nikolaos Tziolas 《manuscripta mathematica》2002,108(4):461-482
In order to calculate the multiplicity of an isolated rational curve C on a local complete intersection variety X, i.e. the length of the local ring of the Hilbert Scheme of X at [C], it is important to study infinitesimal neighborhoods of the curve in X. This is equivalent to infinitesimal extensions of ℙ1 by locally free sheaves. In this paper we study infinitesimal extensions of ℙ1, determine their structure and obtain upper and lower bounds for the length of the local rings of their Hilbert schemes at
[ℙ1].
Received: 11 June 2001 / Revised version: 28 January 2002 相似文献
3.
4.
In this paper we study 16 complete intersection K3-fibered Calabi--Yau variety types in biprojective space ℙ
n
1}×ℙ1. These are all the CICY-types that are K3 fibered by the projection on the second factor. We prove existence of isolated rational curves of bidegree (d,0) for every positive integer d on a general Calabi–Yau variety of these types. The proof depends heavily on existence theorems for curves on K3-surfaces proved by S. Mori and K. Oguiso. Some of these varieties are related to Calabi–Yau varieties in projective space
by a determinantal contraction, and we use this to prove existence of rational curves of every degree for a general Calabi–Yau
variety in projective space.
Received: 14 October 1997 / Revised version: 18 January 1998 相似文献
5.
ABSTRACT. We prove the following form of the Clemens conjecture in low degree. Let d ≤ 9, and let F be a general quintic threefold in P 4. Then (1) the Hilbert scheme of rational, smooth and irreducible curves of degree d on F is finite, nonempty, and reduced; moreover, each curve is embedded in F with normal bundle (?1) ⊕ (?1), and in P 4 with maximal rank. (2) On F, there are no rational, singular, reduced and irreducible curves of degree d, except for the 17,601,000 six-nodal plane quintics (found by Vainsencher). (3) On F, there are no connected, reduced and reducible curves of degree d with rational components. 相似文献
6.
This note deals with the orthogonality between sequences of random variables. The main idea of the note is to apply the results
on equidistant systems of points in a Hilbert space to the case of the space L
2(Ω, F, ℙ) of real square integrable random variables. The main result gives a necessary and sufficient condition for a particular
sequence of random variables (elements of which are taken from sets of equidistant elements of L
2(Ω, F, ℙ) to be orthogonal to some other sequence in L
2(Ω, F, ℙ). The result obtained is interesting from the point of view of the time series analysis, since it can be applied to a
class of sequences random variables that exhibit a monotonically increasing variance. An application to ergodic theorem is
also provided. 相似文献
7.
Every complex projective space of odd dimension carries a natural contact structure. We give first steps towards the enumeration
of curves in ℙ3 tangent to the contact structure. Such a curve is involutive in the sense that its homogeneous ideal is closed under Poisson
bracket. Involutive curves in ℙ3 contained in a plane split as a union of concurrent lines. We give a formula for the number of plane involutive curves of
a given degree in ℙ3 meeting the appropriate number of lines. We also discuss strategies to deal with the enumeration of involutive rational curves. 相似文献
8.
Hurwitz numbers count genus g, degree d covers of ℙ1 with fixed branch locus. This equals the degree of a natural branch map defined on the Hurwitz space. In tropical geometry, algebraic curves are replaced by certain piece-wise linear objects called tropical curves. This paper develops a tropical
counterpart of the branch map and shows that its degree recovers classical Hurwitz numbers. Further, the combinatorial techniques
developed are applied to recover results of Goulden et al. (in Adv. Math. 198:43–92, 2005) and Shadrin et al. (in Adv. Math. 217(1):79–96, 2008) on the piecewise polynomial structure of double Hurwitz numbers in genus 0. 相似文献
9.
For a (smooth irreducible) curveC of genus g and Clifford indexc>2 with a linear seriesg
d
r
computing c (so
) it is well known thatc + 2 ≤d ≤2 (c + 2), and if
then 2c + 1 ≤g ≤ 2c + 4 unlessd = 2c + 4 in which caseg = 2c + 5.
Let c ≥ 0 andg be integers. If 2c + 1 ≤g ≤2c + 4 we prove that for any integerd <g such thatd ≡c mod 2 andc + 2 ≤d < 2(c + 2) there exists a curve of genus g and Clifford index c with a gd
r computing c. Ford ≥c + 6 (i.e.r ≥ 3) we construct this curve on a surface of degree 2r-2 in ℙr, and ford ≥c + 8 (i.e.r ≥ 4) we show that such a curve cannot be found on a surface in ℙr of smaller degree. In fact, if gd
r computes the Clifford index c of C such thatc + 8 ≤d ≤ 2c + 3 then the birational morphism defined by this series cannot map C onto a (maybe, singular) curve contained in a surface
of degree at most 2r-3 in ℙr. 相似文献
10.
Let ƒ be a polynomial automorphism of ℂk of degree λ, whose rational extension to ℙk maps the hyperplane at infinity to a single point. Given any positive closed current S on ℙk of bidegree (1,1), we show that the sequence λ−n(ƒn)*S converges in the sense of currents on ℙk to a linear combination of the Green current T+ of ƒ and the current of integration along the hyperplane at infinity. We give an interpretation of the coefficients in terms
of generalized Lelong numbers with respect to an invariant dynamical current for ƒ−1. 相似文献
11.
Markus Brodmann 《manuscripta mathematica》1992,76(1):181-192
Let M be a generalized Cohen-Macaulay module over a noetherian local ring (R,m). Fix a standard system x1, …, xd∈m with respect to M and let
. We construct a coherent Cohen-Macaulay sheafK over the projective space ℙ
R/I
d-1
whose cohomological Hilbert functions depend only on the lengths of the local cohomology modules H
m
i
(M), (i=0, …, d−1). 相似文献
12.
Let 𝒞⊆ℙ
r
K
be a non-degenerate projective curve of degree d>r+1 of maximal regularity so that 𝒞 has an extremal secant line . We show that 𝒞∪ is arithmetically Cohen Macaulay if d<2r−1 and we study the Betti numbers and the Hartshorne-Rao module of the curve 𝒞.
Received: 27 March 2002; in final form: 24 May 2002 /
Published online: 1 April 2003
Mathematics Subject Classification (1991): 14H45, 13D02.
The second author was partially supported by Swiss National Science Foundation (Projects No. 20-52762.97 and 20-59237.99). 相似文献
13.
Here we investigate the rational cohomology of the moduli space ̄0,n(r,d) of degree d stable maps from n-pointed rational curves to r. We obtain partial results for small values of d with an inductive method inspired by a paper of Enrico Arbarello and Maurizio Cornalba. 相似文献
14.
Luisella Caire 《manuscripta mathematica》1990,67(1):433-450
We generalize and make rigorous a construction by Enriques which allows one to obtain a plane curve as the projection of a
non singular curve spanning ℙ4 we show that every non singular curve in ℙr projecting onto a given plane curve can be obtained by the same construction. Finally we prove that every non singular plane
curve of degree d is the projection of a (non singular) curve of degree 2d-1 spanning ℙ4, and that no lower degree is possible.
Supported by the M. P. I. of the Italian Government 相似文献
15.
We show that the set of the homogeneous saturated ideals with given initial ideal in a fixed term-ordering is locally closed
in the Hilbert scheme, and that it is affine if the initial ideal is saturated. Then, Hilbert schemes can be stratified using
these subschemes. We investigate the behaviour of this stratification with respect to some properties of the closed points.
As application, we describe the singular locus of the component of Hilb4
z
+1
ℙ4 containing the ACM curves of degree 4.
Received: 30 November 1998 / Revised version: 16 September 1999 相似文献
16.
Carlo Magagna 《Monatshefte für Mathematik》2008,153(1):59-81
For a positive integer N, we define the N-rank of a non singular integer d × d matrix A to be the maximum integer r such that there exists a minor of order r whose determinant is not divisible by N. Given a positive integer r, we study the growth of the minimum integer k, such that A
k
− I has N-rank at most r, as a function of N. We show that this integer k goes to infinity faster than log N if and only if for every eigenvalue λ which is not a root of unity, the sum of the dimensions of the eigenspaces relative
to eigenvalues which are multiplicatively dependent with λ and are not roots of unity, plus the dimensions of the eigenspaces
relative to eigenvalues which are roots of unity, does not exceed d − r − 1. This result will be applied to recover a recent theorem of Luca and Shparlinski [6] which states that the group of rational
points of an ordinary elliptic curve E over a finite field with q
n
elements is almost cyclic, in a sense to be defined, when n goes to infinity. We will also extend this result to the product of two elliptic curves over a finite field and show that
the orders of the groups of
rational points of two non isogenous elliptic curves are almost coprime when n approaches infinity.
Author’s address: Dipartimento di Matematica e Informatica, Via Delle Scienze 206, 33100 Udine, Italy 相似文献
17.
Effective algebraic degeneracy 总被引:1,自引:0,他引:1
We show that for every smooth projective hypersurface X⊂ℙ
n+1 of degree d and of arbitrary dimension n
≥2, if X is generic, then there exists a proper algebraic subvariety Y
⊊
X such that every nonconstant entire holomorphic curve f
:ℂ→X has image f(ℂ) which lies in Y, as soon as its degree satisfies the effective lower bound
d\geqslant 2n5d\geqslant 2^{n^{5}}
. 相似文献
18.
Carlo Magagna 《Monatshefte für Mathematik》2008,23(2):59-81
For a positive integer N, we define the N-rank of a non singular integer d × d matrix A to be the maximum integer r such that there exists a minor of order r whose determinant is not divisible by N. Given a positive integer r, we study the growth of the minimum integer k, such that A
k
− I has N-rank at most r, as a function of N. We show that this integer k goes to infinity faster than log N if and only if for every eigenvalue λ which is not a root of unity, the sum of the dimensions of the eigenspaces relative
to eigenvalues which are multiplicatively dependent with λ and are not roots of unity, plus the dimensions of the eigenspaces
relative to eigenvalues which are roots of unity, does not exceed d − r − 1. This result will be applied to recover a recent theorem of Luca and Shparlinski [6] which states that the group of rational
points of an ordinary elliptic curve E over a finite field with q
n
elements is almost cyclic, in a sense to be defined, when n goes to infinity. We will also extend this result to the product of two elliptic curves over a finite field and show that
the orders of the groups of
\Bbb Fqn-{\Bbb F}_{q^n}-
rational points of two non isogenous elliptic curves are almost coprime when n approaches infinity. 相似文献
19.
Gerhard Frey 《Israel Journal of Mathematics》1994,85(1-3):79-83
Thed-th symmetric productC
(d) of a curveC defined over a fieldK is closely related to the set of points ofC of degree ≤d. IfK is a number field, then a conjecture of Lang [Hi] proved by Faltings [Fa2] implies ifC
(d)
(K) is an infinite set, then there is aK-rational covering ofC → ℙ
|K
1
of degree ≤2d. As an application one gets that for fixed fieldK and fixedd there are only finitely many primes ι such that the set of all elliptic curves defined over some extensionsL ofK with [L∶K]≤d and withL-rational isogeny of degree ι is infinite. 相似文献
20.
Romain Dujardin 《Mathematische Annalen》2003,325(4):745-765
In this paper we study laminar currents in ℙ2. Given a sequence of irreducible algebraic curves (C
n
) converging in the sense of currents to T, we find geometric conditions on the curves ensuring that the limit current T is laminar. This criterion is then applied to meromorphic dynamical systems in ℙ2, and laminarity of the dynamical ``Green' current is obtained for a wide class of meromorphic self maps of ℙ2, as well as for all bimeromorphic maps of projective surfaces.
Received: 24 September 2001 / Published online: 10 February 2003
Mathematics Subject Classification (2000): 32U40, 37Fxx, 32H50 相似文献