We prove a multiple-points higher-jets nonvanishing theorem by the use of local Seshadri constants. Applications are given to effectivity problems such as constructing rational and birational maps into Grassmannians, and the global generation of vector bundles.
Let be polynomials in variables without a common zero. Hilbert’s Nullstellensatz says that there are polynomials such that . The effective versions of this result bound the degrees of the in terms of the degrees of the . The aim of this paper is to generalize this to the case when the are replaced by arbitrary ideals. Applications to the Bézout theorem, to Łojasiewicz–type inequalities and to deformation theory are also discussed.
Let p be a prime number, p ≠ 2,3 and Fp the finite field with p elements. An elliptic curve E over Fp is a projective nonsingular curve of genus 1 defined over Fp. Each one of these curves has an isomorphic model given by an (Weierstrass) equation E: y2 = x3 + Ax + B, A,B ∈ Fp with D = 4A3 + 27B2 ≠ 0. The j-invariant of E is defined by j(E) = 1728·4A3/D.The aim of this note is to establish some results concerning the cardinality of the group of points on elliptic curves over Fp with j-invariants...
Continuing our work on the fundamental groups of conic-line arrangements (Amram et al., 2003), we obtain presentations of fundamental groups of the complements of three families of quadric arrangements in P2. The first arrangement is a union of n conics, which are tangent to each other at two common points. The second arrangement is composed of n quadrics which are tangent to each other at one common point. The third arrangement is composed of n quadrics, n-1 of them are tangent to the n-th one...
We consider the Gaudin model associated to a point z ∈ ℂⁿ with pairwise distinct coordinates and to the subspace of singular vectors of a given weight in the tensor product of irreducible finite-dimensional sl₂-representations, [G]. The Bethe equations of this model provide the critical point system of a remarkable rational symmetric function. Any critical orbit determines a common eigenvector of the Gaudin hamiltonians called a Bethe vector. In [ReV], it was shown that for generic...
We present bounds for the degree and the height of the polynomials arising in some problems in effective algebraic geometry including the implicitization of rational maps and the effective Nullstellensatz over a variety. Our treatment is based on arithmetic intersection theory in products of projective spaces and extends to the arithmetic setting constructions and results due to Jelonek. A key role is played by the notion of canonical mixed height of a multiprojective variety. We study this notion...
We prove that Hori-Vafa mirror models for smooth Fano complete intersections in weighted projective spaces admit an interpretation as Laurent polynomials.
We determine all anticanonically embedded quasi smooth log del Pezzo surfaces in weighted projective 3-spaces. Many of these admit a Kähler-Einstein metric and most of them do not have tigers.
We propose a combinatorial method of proving non-specialty of a linear system of curves with multiple points in general position. As an application, we obtain a classification of special linear systems on ℙ¹×ℙ¹ with multiplicities not exceeding 3.
The Cox ring provides a coordinate system on a toric variety analogous to the homogeneous coordinate ring of projective space. Rational maps between projective spaces are described using polynomials in the coordinate ring, and we generalise this to toric varieties, providing a unified description of arbitrary rational maps between toric varieties in terms of their Cox coordinates. Introducing formal roots of polynomials is necessary even in the simplest examples.