Spectral curves of operators with elliptic coefficients.

Eilbeck, J.Chris; Enolski, Victor Z.; Previato, Emma

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Linearly Normal Curves in P^n

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2000 Mathematics Subject Classification: 14H45, 14H50, 14J26. We construct linearly normal curves covering a big range from P^n, n ≥ 6 (Theorems 1.7, 1.9). The problem of existence of such algebraic curves in P^3 has been solved in [4], and extended to P^4 and P^5 in [10]. In both these papers is used the idea appearing in [4] and consisting in adding hyperplane sections to the curves constructed in [6] (for P^3) and [15, 11] (for P^4 and P^5) on some special surfaces. In...

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It is known that in the case of hyperelliptic curves the Shafarevich conjecture can be made effective, i.e., for any number field $k$ and any finite set of places $S$ of $k$ , one can effectively compute the set of isomorphism classes of hyperelliptic curves over $k$ with good reduction outside $S$ . We show here that an extension of this result to an effective Shafarevich conjecture for of hyperelliptic curves of genus $g$ would imply an effective version of Siegel’s theorem for integral points...

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In the present text we give a geometric interpretation of quasi-modular forms using moduli of elliptic curves with marked elements in their de Rham cohomologies. In this way differential equations of modular and quasi-modular forms are interpreted as vector fields on such moduli spaces and they can be calculated from the Gauss-Manin connection of the corresponding universal family of elliptic curves. For the full modular group such a differential equation is calculated and it turns out...

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