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This paper investigates the arithmetic of fractional ideals of a purely cubic function field and the infrastructure of the principal ideal class when the field has unit rank one. First, we describe how irreducible polynomials decompose into prime ideals in the maximal order of the field. We go on to compute so-called canonical bases of ideals; such bases are very suitable for computation. We state algorithms for ideal multiplication and, in the case of unit rank one and characteristic at least five,...
Let be a desingularization of a normal surface . The group Pic is provided with an order relation , defined by . for any effective exceptional divisor . Comparing to the usual order relation we define the ceiling of which is an exceptional divisor. This notion allows us to improve the usual vanishing theorem and we deduce from it a numerical criterion for rationality and a genus formula for a curve on a normal surface; the difficulty lies in the case of a Weil divisor which is not a Cartier...
We study the possible dimension vectors of indecomposable parabolic bundles on the projective line, and use our answer to solve the problem of characterizing those collections of conjugacy classes of n×n matrices for which one can find matrices in their closures whose product is equal to the identity matrix. Both answers depend on the root system of a Kac-Moody Lie algebra. Our proofs use Ringel’s theory of tubular algebras, work of Mihai on the existence of logarithmic connections, the Riemann-Hilbert...
If E is an elliptic curve defined over a quadratic field K, and the j-invariant of E is not 0 or 1728, then has infinite rank. If E is an elliptic curve in Legendre form, y² = x(x-1)(x-λ), where ℚ(λ) is a cubic field, then has infinite rank. If λ ∈ K has a minimal polynomial P(x) of degree 4 and v² = P(u) is an elliptic curve of positive rank over ℚ, we prove that y² = x(x-1)(x-λ) has infinite rank over .
We prove that every injective endomorphism of an affine algebraic variety over an algebraically closed field of characteristic zero is an automorphism. We also construct an analytic curve in ℂ⁶ and its holomorphic bijection which is not a biholomorphism.
We use the methods that were developed by Adler and van Moerbeke to determine explicit
equations for a certain moduli space, that was studied by Narasimhan and Ramanan. Stated
briefly it is, for a fixed non-hyperelliptic Riemann surface of genus , the
moduli space of semi-stable rank two bundles with trivial determinant on . They
showed that it can be realized as a projective variety, more precisely as a quartic
hypersurface of , whose singular locus is the Kummer variety of . We
first construct...
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