Complete Linear System Research Articles

Field approach to the three-dimensional boundary integral formulation

This paper addresses the problem of time-harmonic electromagnetic three-dimensional scattering in dissipative or dielectric media. A field approach to the well-known boundary integral formulation is derived from a general identity between two distinct electromagnetic states. The complete linear system of equations so obtained is overdetermined. A simple analysis of the regularity properties of the subsystems involved is shown.

On the vanishing of higher syzygies of curves

The present paper is related to a conjecture made by Green and Lazarsfeld concerning 1-linear syzygies of curves embedded by complete linear systems of sufficiently large degrees. Given a smooth, irreducible, complex, projective curve $X$, we prove that the least integer $q$ for which the property $(M_q)$ fails for a line bundle $L$ on $X$ does not depend on $L$ as soon as its degree becomes sufficiently large. Consequently, this number is an invariant of the curve, and the statement of Green-Lazarsfeld's conjecture is equivalent to saying that this invariant equals the gonality of the curve. We verify the conjecture for plane curves, curves lying on Hirzebruch surfaces, and for generic curves having the genus sufficiently large compared to the gonality. We conclude the paper by proving that Green's canonical conjecture holds for curves lying on Hirzebruch surfaces.

A bound on the Castelnuovo-Mumford regularity for curves

Recall that a projective curve in \(\Bbb P^r\) with ideal sheaf \(\Cal I\) is said to be n-regular if \(H^i(\Bbb P^r, \Cal I\otimes\Cal O_{\Bbb P^r}(n-i))=0\) for every integer \(i>0\) and that in this case, it is cut out scheme-theoretically by equations of degree at most n. The purpose here is to show that an irreducible, reduced, projective curve of degree d and large arithmetic genus \(p_a\) satisfies a smaller regularity bound than the optimal one \(d-r+2\). For example, if \(p_a\geq r-2\) then a curve is \((d-2r+4)\)-regular unless it is embedded by a complete linear system of degree \(2p_a+2\).

Canonical stability in terms of singularity index for algebraic threefolds

Throughout the ground field is always supposed to be algebraically closed of characteristic zero. Let X be a smooth projective threefold of general type, denote by ϕm the m-canonical map of X which is nothing but the rational map naturally associated with the complete linear system [mid ]mKX[mid ]. Since, once given such a 3-fold X, ϕm is birational whenever m [Gt ] 0, quite an interesting thing to find is the optimal bound for such an m. This bound is important because it is not only crucial to the classification theory, but also strongly related to other problems. For example, it can be applied to determine the order of the birational automorphism group of X [21, remark in section 1]. To fix the terminology we say that ϕm is stably birational if ϕt is birational onto its image for all t [ges ] m. It is well known that the parallel problem in the surface case was solved by Bombieri [1] and others. In the 3-dimensional case, many authors have studied the problem, in quite different ways. Because, in this paper, we are interested in the results obtained by Hanamura [7], we do not plan to mention more references here. According to 3-dimensional MMP, X has a minimal model which is a normal projective 3-fold with only ℚ-factorial terminal singularities. Though X may have many minimal models, the singularity index (namely the canonical index) of any of its minimal models is uniquely determined by X. Denote by r the canonical index of minimal models of X. When r = 1 we know that ϕ6 is stably birational by virtue of [3, 6, 13 and 14]. When r [ges ] 2, Hanamura proved the following theorem.

LINE BUNDLES OF TYPE (1,…,1,2,…,2,4,…,4) ON ABELIAN VARIETIES

We show birationality of the morphism associated to line bundles L of type (1,…,1,2,…,2,4,…,4) on a generic g-dimensional abelian variety into its complete linear system such that h0(L) = 2g. When g = 3, we describe the image of the abelian threefold and from the geometry of the moduli space SUC(2) in the linear system |2θC|, we obtain analogous results in ℙH0(L).

On the geometry of algebraic curves having many real components

We show that there is a large class of nonspecial effective divisors of relatively small degree on real algebraic curves having many real components i.e. on M-curves. We apply to 1. complete linear systems onM-curves containing divisors with entirely real support, and 2. morphisms of M-curves into P1.

A tight closure proof of Fujita's freeness conjecture for very ample line bundles

The proof here actually proves a stronger statement than Theorem 1 above. The variety X need not be smooth; F-rationality is sufficient (the definition is recalled in the next section; in characteristic zero it is equivalent to rational singularities). Also, the line bundle L need not be very ample; it is sufficient if L is globally generated and the dimension the complete linear system |L| is greater than d. These generalizations are summarized in Theorem 2. Furthermore, with a little more work, the same ideas prove even stronger statements, which are interesting algebraically, but difficult to interpret geometrically (see Theorem 3). Fujita’s Freeness Conjecture predicts the same conclusion under the much weaker hypothesis that L is only ample. While open in general, for varieties defined over a field of characteristic zero, it is known in dimension four or less [R], [EL], [Ka]. See also [AS] for important progress on the conjecture, and [Ko] for a good survey about it. In characteristic zero, it is not hard to give a geometric argument of the special case above using the Kodaira Vanishing Theorem. The goal here is to give a simple, quite different proof that is purely algebraic and valid in any characteristic. This argument offers a nice illustration of how tight closure can be used to prove geometric theorems in arbitrary characteristic without the use of the usual tools of desingularization or vanishing theorems.

Syzygies of abelian varieties

Let A be an ample line bundle on an abelian variety X (over an algebraically closed field). A theorem of Koizumi ([Ko], [S]), developing Mumford's ideas and results ([M]), states that if m > 3 the line bundle L = AO' embeds X in projective space as a projectively normal variety. Moreover, a celebrated theorem of Mumford ([M2]), slightly refined by Kempf ([K4]), asserts that the homogeneous ideal of X is generated by quadrics as soon as m > 4. Such results turn out to be particular cases of a statement, conjectured by Rob Lazarsfeld, concerning the minimal resolution of the graded algebra RL = 0% Ho(X, L?h) over the polynomial ring SL DO SymhHO (X, L). The purpose of this paper is to prove Lazarsfeld's conjecture. To put such matters into perspective, it is useful to review the case of projective curves. A classical theorem of Castelnuovo states that a curve X, embedded in projective space by a complete linear system ILI, is projectively normal as soon as deg L > 2g(X) + 1, and a theorem of Mattuck, Fujita and Saint-Donat states that if deg L > 2g(X) + 2, then the homogeneous ideal of X is generated by quadrics. Green ([G1]) unified, re-interpreted and generalized these results to a statement about syzygies. Specifically, given a (smooth) projective variety X and a very ample line bundle L on X, a minimal resolution of RL as a graded SL-module (notation as above) looks like

ON THE DEGREE AND THE BIRATIONALITY OF THE SECOND ADJUNCTION MAPPING

Let ℒ be a very ample line bundle on ℳ, a projective manifold of dimension n ≥3. Under the assumption that Kℳ + (n-2) ℒ has Kodaira dimension n, we study the degree of the map ϕ associated to the complete linear system |2(KM + (n-2) L)|, where (M, L) is the first reduction of (ℳ, ℒ). In particular we show that under a number of conditions, e.g. n ≥ 5 or Kℳ + (n-3)ℒ having nonnegative Kodaira dimension, the degree of ϕ is one, i.e. ϕ is birational. We also show that under a mild condition on the linear system |KM + (n-2) L| satisfied for all known examples, ϕ is birational unless (ℳ, ℒ) is a three dimensional variety with very restricted invariants. Moreover there is an example with these invariants such that deg ϕ= 2.

A base point free theorem of Reid type, II

Let $X$ be a complete algebraic variety over $\mathbf{C}$. We consider a log variety $(X, \Delta)$ that is weakly Kawamata log terminal. We assume that $K_X + \Delta$ is a $\mathbf{Q}$-Cartier $\mathbf{Q}$-divisor and that every irreducible component of $\lfloor \Delta \rfloor$ is $\mathbf{Q}$-Cartier. A nef and big $\mathbf{Q}$-Cartier $\mathbf{Q}$-divisor $H$ on $X$ is called nef and log big on $(X, \Delta)$ if $H \vert_B$ is nef and big for every center $B$ of non-``Kawamata log terminal'' singularities for $(X, \Delta)$. We prove that, if $L$ is a nef Cartier divisor such that $aL - (K_X + \Delta)$ is nef and log big on $(X, \Delta)$ for some $a \in \mathbf{N}$, then the complete linear system $|mL|$ is base point free for $m \gg 0$.

Existence of pencils with prescribed scrollar invariants of some general type

Let C be an irreducible smooth projective non-hyperelliptic curve of genus defined over the field C of complex numbers. Let g be a complete base-point free special linear system on C. The scrollar invariants of g are defined as follows. Let C be canonically embedded in P~ and let X be the union of the linear spans (D) with D g . This defines a set of integers e > ... > ek-i > 0 such that X is the image of the projective bundle P(e\\... e^-i) = P(OPι (ei) Θ . . . Θ OPι (βk-i)) using the tautological bundle (see e.g. [2]; [7]). Those integers ι ;e 2 ; . . . e^-i are called the scrollar invariants of g\. Those scrollar invariants determine (and are determined by) the complete linear systems associated to multiples of the linear system g\. For 1 i Here KC denotes a canonical divisor on C. Let m = βfc-ι+2. Then m is defined by the following conditions: dim(|(ra—l)p£|) — m—1 and dim(|ra ra. In case \mg is birationally very ample then the scrollar invariants satisfy the inequalities e* '^'~^m 1 -f (x — (j — l)m + l) j . Equality (if not in conflict with the Riemann-Roch Theorem) can be expected being the most general case for a fixed value of m. The inequalities also imply dim(|(λ; — l)mg\.\) = dim(|((fc — l)ra 1)^ | ) 4k. This implies that \((k — l)m l)g is not special. Using the dimension bound one obtains < [(k k)m 2k + 2]/2. (This easy but interesting consequence from the inequalities is not mentioned by Kato and Ohbuchi.) In this paper we prove the following theorem.

Une caractérisation différentielle des points de weierstraß généralisés d'une surface de riemann compacte de genre g ≥ 2

We show that the set S[ n] of the Weierstraß points of order n, associated to the linear system | nK S |, where K S is a canonical divisor on a compact Riemann surface S of genus g ≥ 2, coincides with the zeros of the canonical representant of the first Chern class of a holomorphic line bundle, Hermitian for a metric induced by the canonical hermitian jacobian metric on S. This result extends to the case of the Weierstraß points associated to a complete linear system without fixed points.

Prediction of Strip Profile in Rolling Process Using Influence Coefficients and Boussinesq’s Equations

A simplified numerical method is proposed in this article to predict strip profile. The mill is simulated by beams (rolls) on the infinite elastic half space (between rolls and strip). The roll deflection and indentation are calculated using the influence coefficients and Boussinesq’s equations respectively. The roll contours in the contact zone can be expressed in terms of the influence coefficient and indentation matrices. Compatibility equations are derived by matching two surface profiles in the contact zone. Equilibrium equations ensure force and moment balance of the strip and each roll. A complete linear system is established by assembling compatibility and equilibrium equations. The contact rolling pressure distribution and the rigid motion of the roll chocks can be solved. Roll surface contours and strip profiles can be further calculated accordingly.

Nodal curves on surfaces of general type

plane irreducible curves of degree d having exactly nodes and no other singularities is non empty and everywhere smooth of codimension in the linear system |O(d)|. If C ∈Vd; Severi uses the non speciality of the normal line bundle to the composition : C → C → P, where C is the normalization of C, to prove that Vd; is smooth of the asserted codimension at the point C. This proof can be extended to rational, ruled and K3 surfaces with little changes: we discuss this point in Sect. 1 (see also [T] for the case of rational surfaces). In the case of a surface of general type S the approach of Severi fails, and in fact it is easy to see that the analogous of Severi’s theorem does not hold in general if we impose too many nodes to the curves of a complete linear system |D|. One may nevertheless look for an upper bound on ensuring that the family VD; of irreducible curves in |D| with nodes is smooth of codimension . In Sect. 2 we give the following partial answer to this problem:

On syzygies of projective varieties

Let X be a projective variety. The conditions that guarantee projective normality and quadratic generations of the ideal defining the embedding of X were studied classically. These results of Mumford et al. were considered by Mark Green as the first step towards understanding the higher syzygies. We say that a line bundle L on X satisfies Property Np if the ideal defining the embedding of X in the complete linear system of L is generated by quadratic equations and has linear syzygies till p stage (see [2]). In [3], Green introduced the notion of Koszul cohomology and proved that vanishing of certain higher Koszul cohomology is equivalent to the Property Np. In [4], it was proved that the required higher Koszul cohomology groups vanish for sufficiently ample line bundles ([4, Theorem 3.2]). However, the proof given in [4] makes a tacit assumption that the ideal sheaf of the diagonal embedding of X in its two-fold product X is locally free which is not true. If X is a smooth projective variety, using different methods, L. Ein and R. Lazarsfeld obtained an effective bound on the power of L required for satisfying Property Np (see [2]). In this article, we give a criterion for a line bundle L to have Property Np in terms of vanishing of certain first cohomology. Our method does not assume the smoothness of X. Using this criterion, we see that sufficiently high power of L will have property Np for fixed p. Therefore, we also see that higher Koszul cohomology of a sufficiently high power of an ample line bundle vanish. This answers the question 5.13 of [3]. Here, we would like to mention that when X is a smooth variety, a vanishing theorem of M. Nori can also be used to answer this question ([8, Proposition 3.4]).

On the minimal free resolution of some projective curves

Here we study the minimal free resolution of non linearly normal embedded curves of high degree and the homogeneous ideal of general enough curves (for general k-gonal curves and, in a much under range, for curves with general moduli). Then we consider the surjectivity of the Gaussian maps (or «Wahl maps») for non complete linear systems.

Frobenius Splitting of Schubert Varieties and Linear Syzygies

Introduction. Let G be a connected, simply connected, semi-simple alge? braic group. Let B be a Borel subgroup of G. For an ample line bundle L on G/B, the questions about projective normality of the embedding of G/B in the complete linear system of L and the equations defining the embedding were extensively studied. Ramanathan in [13] proved that the ideal defining the embedding of G/B in the complete linear system of L is generated by quadratic equations. Kempf and Ramanathan proved that the equations defining the embedding of G/B in the complete linear system of L\,... ,Lr are quadratic and that a Schubert variety X is linearly defined with respect to L\,... ,Lr. Ramanathan had conjectured that the method of Frobenius splitting (used in [13]) can be used to study higher syzygies. Let R (BH?(G/B,?Ljj), where the vector (mj) ranges in Nr, be the multigraded coordinate ring of G/B with respect to L\,... ,Lr. In this paper, we use Frobenius splittings to study the higher syzygies of the irrelevant maximal ideal m of R. We prove that m as a /?-module has linear syzygies. Our methods also give simpler proofs of the results obtained in [6] and [13]. Let Z be a projective variety defined over an algebraically closed field. Let L\,... ,L, be ample line bundles on Z and let m denote the irrelevant maximal ideal of the multi-graded ring R = @H?(Z,?Ljj). In the first section we give a criterion for m to have linear syzigies over R. We show that if for every n > 2, the first cohomology of the n-fold product Zn with values in certain ideal sheaves tensored with powers of Lj vanishes then m has linear syzygies (Propositions 1.9 and 1.10). \ye also give a similar criterion for a homogeneous quotient ring of R to have linear syzygies over R (Proposition 1.13). In the second section, we show that any r ample line bundles L\,...,L, on G/B, and their restrictions to a Schubert variety X C G/B over an algebracally closed field of positive characteristic satisfy the above criteria. To prove this result, we use the method of Frobenius splittings. If Z is a Frobenius split variety, and if L is an ample line bundle on Z, then we have Hl(Z,L) = 0 for / > 1. Further if Y is a compatibly split subvariety of Z, then the restriction

Non-trivial Linear Systems on Smooth Plane Curves

Let $C$ be a smooth plane curve of degree $d$ defined over an algebraically closed field $k$. A base point free complete very special linear system $g^r_n$ on $C$ is trivial if there exists an integer $m\ge 0$ and an effective divisor $E$ on $C$ of degree $md-n$ such that $g^r_n=|mg^2_d-E|$ and $r=(m^2+3m)/2-(md-n)$. In this paper, we prove the following: Theorem Let $g^r_n$ be a base point free very special non-trivial complete linear system on $C$. Write $r=(x+1)(x+2)/2-b$ with $x, b$ integers satisfying $x\ge 1, 0\le b \le x$. Then $n\ge n(r):=(d-3)(x+3)-b$. Moreover, this inequality is best possible.

On the general osculating flag of a projection of a curve in char. P

This paper study properties (in positive caracteristic) of the general osculating flag to a projective curve embedded inP n by a non complete linear system of high degree.

Noether's Inequality for Non-complete Algebraic Surfaces of General Type

Let V be a nonsingular projective surface of Kodaira dimension K(V) > 0. Let D be a reduced, effective, nonzero divisor on V with only normal crossings. In the present article, a pair (F, D) is said to be a minimal logarithmic surface of general type, if, by definition, Kv + D is a numerically effective divisor of self intersection number (Kv + D) 2 > 0 and if Kv + D has positive intersection with every exceptional curve of the first kind on V. Here Kv is the canonical divisor of V. In the case, on the one hand, Sakai [8; Theorem 7.6] proved a Miyaoka — Yau type inequality (cf ) := (Kv + D) 2 — c2 -by making use of [8; Theorem 5.5]. In the present article, we shall prove that (cf ) > -c2 — 2 provided that the rational map &\Ky+D defined by the complete linear system \KV + D| has a surface as the image of V. Moreover, if the equality holds, then the logarithmic geometric genus pg := h°(V, Kv + D) = -(cf ) + 2 = 3, D is an elliptic curve and V is the canonical resolution in the sense of Horikawa associated with a double covering h: Y -> P. In addition, the branch locus B of h is a reduced curve of degree eight and the singular locus Sing B consists of points of multiplicity < 3 except for at most one simple quadruple point. Introduction This is a succession of the previous paper [9]. We work over an algebraically closed field fe of characteristic zero. Let V be a nonsingular projective surface defined over fe. If V is a minimal surface of general type, we have the following inequality due to M. Noether: This inequality, together with the Noether formula 12%((9V) = c±(V) 2 + c2(V Communicated by K. Saito, May 18, 1990. 1991 Mathematics Subject Classification: 14J29 Department of Mathematics, the National University of Singapore, Singapore.