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Linear Codes on Nonsingular Curves are Better than Those on Singular Curves
, 1999
"... this paper, we show that for fixed designed minimum distance in a widerange, the dimension of codes on a singular curve is smaller than or equal to that of the codes on itsnormalization, and the number of check symbols of the former codes is larger than that of the latter codes. This implies the opt ..."
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the optimality of nonsingular curves for code construction.
Real cohomology groups of the space of nonsingular curves of degree 5 in CP 2
, 2008
"... Let us denote by Π5 the space of all homogeneous polynomials C 3 → C of degree 5 and by P5 its subspace consisting of all nonsingular polynomials (i.e. the polynomials, whose gradient is nonzero outside the origin). Theorem 1 The Poincaré polynomial of P5 is equal to (1 + t)(1 + t 3)(1 + t 5). A ge ..."
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general method of calculating the cohomology of the spaces of nonsingular algebraic hypersurfaces of given degree was described in [2]. In particular, the real cohomology groups of spaces of nonsingular plane curves of degree ≤ 4 were calculated there. In the present work, we apply a modification
Real cohomology groups of the space of nonsingular curves of degree 5 in CP 2
"... Let us denote by Π5 the space of all homogeneous polynomials C 3 → C of degree 5 and by P5 its subspace consisting of all nonsingular polynomials (i.e. the polynomials, whose gradient is nonzero outside the origin). Theorem 1 The Poincaré polynomial of P5 is equal to (1 + t)(1 + t 3)(1 + t 5). A ge ..."
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general method of calculating the cohomology of the spaces of nonsingular algebraic hypersurfaces of given degree was described in [2]. In particular, the real cohomology groups of spaces of nonsingular plane curves of degree ≤ 4 were calculated there. In the present work, we apply a modification
The irreducibility of the space of curves of given genus
 Publ. Math. IHES
, 1969
"... Fix an algebraically closed field k. Let Mg be the moduli space of curves of genus g over k. The main result of this note is that Mg is irreducible for every k. Of course, whether or not M s is irreducible depends only on the characteristic of k. When the characteristic s o, we can assume that k ~ ..."
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is to construct families of curves X, some singular, with pa(X)=g, over nonsingular parameter spaces, which in some sense contain enough singular curves to link together any two components that Mg might have. The essential thing that makes this method work now is a recent " stable reduction theorem "
1. Verlinde bundles 1.1. Flatness constraint. Let Mg be the moduli space of nonsingular curves of genus g ≥ 2. Let
"... be the moduli space of rank r degree d semistable bundles on nonsingular genus g curves. The space Ug(r, r(g − 1)) carries a canonical theta divisor Θr = {(C,E → C) with h 0(C,E) 6 = 0}. For levels k ≥ 1, the divisors Θkr are known to have no higher cohomology on the fibers of µ. The µpushforwards ..."
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be the moduli space of rank r degree d semistable bundles on nonsingular genus g curves. The space Ug(r, r(g − 1)) carries a canonical theta divisor Θr = {(C,E → C) with h 0(C,E) 6 = 0}. For levels k ≥ 1, the divisors Θkr are known to have no higher cohomology on the fibers of µ. The µ
Polynomial parametrization of nonsingular algebraic curves
"... In this paper, we give a new method for deciding whether a given nonsingular algebraic curve in K n, which is an intersection of n−1 hypersurfaces, is polynomially parameterizable, and if so computing such a parametrization. Our method is based on locally nilpotent derivations theory. ..."
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In this paper, we give a new method for deciding whether a given nonsingular algebraic curve in K n, which is an intersection of n−1 hypersurfaces, is polynomially parameterizable, and if so computing such a parametrization. Our method is based on locally nilpotent derivations theory.
Results 1  10
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32,378