On the homogeneous ideal of projectively normal curves

Summary Fix integers k, d, g with g0, dg+3, k>0, 2k<(d–g), d(g(k+1)/k) + k+1. Here we prove that for a general curve X of genus g and a general L Pic^d(X), L is normally presented.     

On the postulation of a general projection of a curve inP 3

Summary We say that a curve C in P ³ has maximal rank if for every integer k the restriction map r_c(k):H ⁰(P ³, O_P3(k)) H⁰ (C, O_C(k))has maximal rank. Here we prove the following results. Theorem 1Fix integers g, d with 0g3,dg+3.Fix a curve X of genus g and L Pic^d (X).If g=3and X is hyperelliptic, assume d8. Let _L(X)be the image of X by the complete linear system H ⁰(X, L). Then a general projection of _L(X)into P ³ has maximal rank. Theorem 2For every integer g0,there exists an integer d(g, 3)such that for every dd(g, 3),for every smooth curve X of genus g and every LPic^d (X) the general projection of _L(X)into P ³ has maximal rank.     

On the postulation of a general projection of a curve inP N,N⩾4

Summary Fix a curve X of genus g and L Pic ^d (X). Let _L(X) be the image of X through the complete linear system H⁰(X, L). Here we prove that a general projection of _L(X) intoP ^N has maximal rank if either (a) N4, 0gN–1, dg+N, or (b) dd (g, N) for suitable d(g, N).     

Even diameters of the classes W(r)H ω in the space C2π

For even values of n we find the exact values of the diameters d_n(W^(r)H) of the classes of 2-periodic functions ((t) is an arbitrary convex upwards modulus of continuity) in the space C₂. We find that d_2n(W^(r)H)=d_2n–1(W^(r)H) (n=1, 2, ... r=0, 1, 2, ...).Translated from Matematicheskie Zametki, Vol. 15, No. 3, pp. 387–392, March, 1974.The author expresses his thanks to N. P. Korneichuk for his interest in my work.     

Linear series computing the Clifford index of a projective curve

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 g_d ^r computing c. Ford ≥c + 6 (i.e.r ≥ 3) we construct this curve on a surface of degree 2r-2 in ℙ^r, and ford ≥c + 8 (i.e.r ≥ 4) we show that such a curve cannot be found on a surface in ℙ^r of smaller degree. In fact, if g_d ^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 ℙ^r.     

Some remarks on the schemesW d r

Summary Let X be an irreducible smooth projective curve of genus g. Let _d ^r (g) be the Brill-Noether Number. In this paper we prove some results concerning the schemes W _d ^r of special divisors. 1) Suppose dim (W _d–1 ^r )= _{d– 1} ^r (g)0 and _d ^r (g) < g. If W _{d– 1} ^r is a reduced (resp. irreducible) scheme, then W _d ^r is a reduced (resp. irreducible) scheme. 2) Under certain conditions, if Z is a generically reduced irreducible component of W _d–1 ^r then Z W ₁ ⁰ is a generically reduced irreducible component of W _d ^r . For r=1, we obtain some further results in this direction. 3) As an application of it we are able to prove some dimension theorems for the schemes W _d ¹ .     

An extension of Jung’s theorem

Theorem. Let a set X?Rⁿ have unit circumradius and let B be the unit ball containing X. Put C =conv \(\bar X\) D =diam C (=diam X), k =dim C,d _i = √(2i + 2)/i. Then: (i) D∈[d_n, 2]; (ii) k≧m where m∈{2,3,...,n} satisfies D∈[d_m, d_m?1) (d_i decreases by i); (iii) In case k=m (by (ii), this is always the case when m=n), C contains a k-simplex Δ such that: (α) its vertices are on δB; (β) the centre of B belongs toint Δ; (γ) the inequalitiesλ _k (D) ≦l ≦D with $$\lambda _k (D) = D\sqrt {\frac{{4k - 2D^2 (k - 1)}}{{2 - (k - 2)(D^2 - 2)}}, D \in (d_k ,d_{k - 1} )} $$ are unimprovable estimates for length l of any edge of Δ.     

Sample path properties of processes with stable components

Summary In this paper, processes in R ^d of the form X(t)=(X ₁ (t), X ₂ (t), , X _N (t), where X _i (t) is a stable process of index _i in Euclidean space of dimension d _i and d=d ₁ + + d _N , are considered. The asymptotic behaviour of the first passage time out of a sphere, and of the sojourn time in a sphere is established. Properties of the space-time process (X ₁ (t), t) in R ^d+1 are obtained when X ₁ (t) is a stable process in R ^d . For each of these processes, a Hausdorff measure function (h) is found such that the range set R(s) of the sample path on [0, s] has Hausdorff -measure c s for a suitable finite positive c.During the preparation of this paper, the first author was supported in part by N. S. F. Grant No GP-3906.     

On a generalization of the Cauchy functional equation

Summary While looking for solutions of some functional equations and systems of functional equations introduced by S. Midura and their generalizations, we came across the problem of solving the equationg(ax + by) = Ag(x) + Bg(y) + L(x, y) (1) in the class of functions mapping a non-empty subsetP of a linear spaceX over a commutative fieldK, satisfying the conditionaP + bP P, into a linear spaceY over a commutative fieldF, whereL: X × X Y is biadditive,a, b K\{0}, andA, B F\{0}. Theorem.Suppose that K is either R or C, F is of characteristic zero, there exist A ₁,A ₂,B ₁,B ₂, F\ {0}with L(ax, y) = A ₁ L(x, y), L(x, ay) = A ₂ L(x, y), L(bx, y) = B ₁ L(x, y), and L(x, by) = B ₂ L(x, y) for x, y X, and P has a non-empty convex and algebraically open subset. Then the functional equation (1)has a solution in the class of functions g: P Y iff the following two conditions hold: L(x, y) = L(y, x) for x, y X, (2)if L 0, then A ₁ =A ₂,B ₁ =B ₂,A = A ₁ ² ,and B = B ₁ ² . (3) Furthermore, if conditions (2)and (3)are valid, then a function g: P Y satisfies the equation (1)iff there exist a y ₀ Y and an additive function h: X Y such that if A + B 1, then y ₀ = 0;h(ax) = Ah(x), h(bx) =Bh(x) for x X; g(x) = h(x) + y ₀ + 1/2A ₁ ^-1 B ₁ ^-1 L(x, x)for x P.     

Milnor K-theory of Smooth Varieties

Let k be a field and X a smooth projective variety of dimension d over k. Generalizing a construction of Kato and Somekawa, we define a Milnor-type group which is isomorphic to the ordinary Milnor We prove that is isomorphic to both the higher Chow group CH^d+s (X,s) and the Zariski cohomology group      

Random polytopes in thed-dimensional cube

LetC ^d be the set of vertices of ad-dimensional cube,C ^d ={(x ₁, ...,x _d ):x _i =±1}. Let us choose a randomn-element subsetA(n) ofC ^d . Here we prove that Prob (the origin belongs to the convA(2d+x2d))=(x)+o(1) ifx is fixed andd . That is, for an arbitrary>0 the convex hull of more than (2+)d vertices almost always contains 0 while the convex hull of less than (2-)d points almost always avoids it.     

The real spectrum of an ideal and KO-theory exact sequences

A construction in abstract real algebra is used to define invariants S _n(A) of commutative rings, with or without identity. If A=C(X) is the ring of continuous real functions on a compact space, then S _n(A) = k0^–n(X), and, for any A, S _n(A) Z[1/2]-W _n(A) Z[1/2], where the W _n(A) are the Witt groups of A. In addition, a short exact sequence of rings yields a long exact sequence of the groups S _n. The functors S _n(A) thus provide a solution of a problem proposed by Karoubi. This paper primarily deals with the exact sequences involving a ring A and an ideal I A. Work supported in part by NSF Grant DMS85-06816.     

Weierstrass multiple points on algebraic curves and ramified coverings

Here we prove the following result on Weierstrass multiple points. Theorem:Fix integers k, g with k≥5 and g>4k. Then there exist a genus g, Riemann surface X and k points P ₁, …,P _k of X such that for all integers b ₁≥…≥b _k ≥0we have:

The p-rank of Artin-Schreier curves

The groundfield k is algebraically closed and of characteristic p O. The p-rank of an abelian variety A/k is _A if there are _A copies of Z/pZ in the group of points of order p in A(k). The p-rank _X of a curve X/k is the p-rank of its Jacobian. In general the genus of X is _X. X is ordinary if equality holds.Proposition 3.2 proves that the Artin-Schreier curve X_p with equation (x^p–x)(y^p–y)=1 is ordinary. As its genus is (p–1)(p–1) and it has at least 2p. p. (p–1) automorphisms, it is an ordinary counter example of Hurwitz's theorem if p>37. Theorem 3.5 is the inductive step in extending this to smaller characteristics. Both are corollaries of Theorem 4.1 which is our principal result: if YX is a cyclic covering of degree p ramified at n distinct points, then (_Y–1+n)=(_X–1+n)×p. The particular case n=0, the unramiried case, is due to afarevi [7].The preparation of this paper was supported by the Memorial University of Newfoundland and NRC Grant A-8777.     

Extensions of an inequality of Bonnesen toD-dimensional space and curvature conditions for convex bodies

For convex bodies inE ^d (d 3) with diameter 2 we consider inequalitiesW _i – W _d–1 +( - 1) W _d 0 (i = 0, , d – 2) whereW _j are the quermassintegrals. In addition, for a ball, equality is attained for a body of revolution for which the elementary symmetric functions _d–1–i of main curvature radii is constant. The inequality is actually proved fori = d – 2 by means of Weierstrass's fundamental theorem of the calculus of variations.Dedicated to Professor Otto Haupt with best wishes on his 100th birthday     

Solution of Hadwiger-Levi's Covering Problem for Duals of Cyclic 2k-Polytopes

For a collection C of convex bodies let h(C) be the minimum number m with the property that every element K of C can be covered by m or fewer smaller homothetic copies of K. Denote by C _d ^* the collection of all duals of cyclic d-polytopes, d 2. We show that h(C _2k ^* )=(k +1)2 for any k 2. We also prove the inequalities (d+1) (d+3)/4 h(C _d ^* ) (d+1) 2/2$ for any d 2.     

G. Martens′ Dimension Theorem for Curves of Odd Gonality

For a smooth projective irreducible algebraic curve C of odd gonality, the maximal possible dimension of the variety of special linear systems W _d ^r (C) is d-3r by a result of M. Coppens et al. Furthermore it is known that if the maximum dimension of W(C) for a curve C of odd gonality is attained then C is of very special type of curves by the recent progress made by G. Martens. The purpose of this paper is to chase an extension of the result of G. Martens; if dim W(C)=d-3r-1 for a curve C of odd gonality for some dg-4 and r1, then C must be either a smooth plane sextic, a pentagonal curve of bounded genus or a smooth plane octic.     

Best polynomial approximation in Sobolev-Laguerre and Sobolev-Legendre spaces

We investigate limiting behavior as γ tends to ∞ of the best polynomial approximations in the Sobolev-Laguerre space W^N,2([0, ∞); e^−x) and the Sobolev-Legendre space W^N,2([−1, 1]) with respect to the Sobolev-Laguerre inner product

On Plane Maximal Curves

The number N of rational points on an algebraic curve of genus g over a finite field satisfies the Hasse–Weil bound . A curve that attains this bound is called maximal. With and , it is known that maximalcurves have . Maximal curves with have been characterized up to isomorphism. A natural genus to be studied is and for this genus there are two non-isomorphic maximal curves known when . Here, a maximal curve with genus g ₂ and a non-singular plane model is characterized as a Fermat curve of degree .     

Об отделимости множе ств гиперплоскостью вL p

LetA and be two arbitrary sets in the real spaceL _p, 1p<. Sufficient conditions are obtained for their strict separability by a hyperplane, in terms of the distance between the setsd(A,B) _p=inf{x-y_p,xA,yB} and their diametersd(A) _p, d(B)_p, whered(A) _p=sup{x-y_p; x,yA}. In particular, it is proved that if in an infinite-demensional spaceL _p we haved ^r(A,B)_p>2^–r+1(d^r(A)_p+d^r(B)_p), r=min{p, p(p–1)^–1}, then there is a hyperplane which separatesA andB. On the other hand, the conditiond ^r(A,B)_p=2^–r+1(d^r(A)_p+d^r(B)_p) does not guarantee strict separability. Earlier these results where obtained by V. L. Dol'nikov for the case of Euclidean spaces.