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Register a new userOn rational maps from the product of two general curves
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This paper treats the dominant rational maps from the product of two very general curves to nonsingular projective surfaces. Combining the result by Bastianelli and Pirola, we prove that the product of two very general curves of genus $ggeq 7$ and $ggeq 3$ does not admit dominant rational maps of degree $> 1$ if the image surface is non-ruled. We also treat the case of the 2-symmetric product of a curve.
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Consider a simple algebraic group G of adjoint type, and its wonderful compactification X. We show that X admits a unique family of minimal rational curves, and we explicitly describe the subfamily consisting of curves through a general point. As an application, we show that X has the target rigidity property when G is not of type A_1 or C.
We provide an algorithm to check whether two rational space curves are related by a similarity. The algorithm exploits the relationship between the curvatures and torsions of two similar curves, which is formulated in a computer algebra setting. Helical curves, where curvature and torsion are proportional, need to be distinguished as a special case. The algorithm is easy to implement, as it involves only standard computer algebra techniques, such as greatest common divisors and resultants, and Grobner basis for the special case of helical curves. Details on the implementation and experimentation carried out using the computer algebra system Maple 18 are provided.
Let $K$ be a number field, let $phi in K(t)$ be a rational map of degree at least 2, and let $alpha, beta in K$. We show that if $alpha$ is not in the forward orbit of $beta$, then there is a positive proportion of primes ${mathfrak p}$ of $K$ such that $alpha mod {mathfrak p}$ is not in the forward orbit of $beta mod {mathfrak p}$. Moreover, we show that a similar result holds for several maps and several points. We also present heuristic and numerical evidence that a higher dimensional analog of this result is unlikely to be true if we replace $alpha$ by a hypersurface, such as the ramification locus of a morphism $phi : {mathbb P}^{n} to {mathbb P}^{n}$.
Let $C$ be an algebraic curve of genus $g$ and $L$ a line bundle over $C$. Let $mathcal{MS}_C(n,L)$ and $mathcal{MO}_C(n,L)$ be the moduli spaces of $L$-valued symplectic and orthogonal bundles respectively, over $C$ of rank $n$. We construct rational curves on these moduli spaces which generalize Hecke curves on the moduli space of vector bundles. As a main result, we show that these Hecke type curves have the minimal degree among the rational curves passing through a general point of the moduli spaces. As its byproducts, we show the non-abelian Torelli theorem and compute the automorphism group of moduli spaces.
In a joint work with N. Mok in 1997, we proved that for an irreducible representation $G subset {bf GL}(V),$ if a holomorphic $G$-structure exists on a uniruled projective manifold, then the Lie algebra of $G$ has nonzero prolongation. We tried to generalize this to an arbitrary connected algebraic subgroup $G subset {bf GL}(V)$ and a complex manifold containing an immersed rational curve, but the proposed proof had a flaw.
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