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Explicit Real Cubic Surfaces

The topological classification of smooth real cubic surfaces is recalled and compared to the classification in terms of the number of real lines and of real tritangent planes, as obtained by L. Schlafli in 1858. Using this, explicit examples of surfaces of every possible type are given.

Some Genus 3 Curves with Many Points

We explain a naive approach towards the problem of finding genus 3 curves C over any given finite field Fq of odd characteristic, with a number of rational points close to the Hasse-Weil-Serre upper bound q +1+3[2?q]. The method turns out to be successful at least in characteristic 3.

A remark on parameterizing nonsingular cubic surfaces

Extending a geometric construction due to Sederberg and to Bajaj, Holt, and Netravali, an algorithm is presented for parameterizing a nonsingular cubic surface by polynomials of degree three. The fact that such a parametrization exists is classical. The present algorithm is, by its purely geometric nature, a very natural one. Moreover, it contains a practical way of finding all lines in an implicitly given cubic surface. Two explicit examples are presented, namely the classical Clebsch diagonal surface and the cubic Fermat surface.

On a conjecture of Sophus Lie

Sophus Lie conjectured, that any finite-dimensional complex transitive Lie algebra of vector fields in the variables x1, . . . , xn has a conjugate of which all coefficients lie in the algebra E generated by the xi and the exponentials exp(λxi). This paper treats this conjecture in the setting of formal power series. First, we derive a formula that realizes an abstractly given transitive Lie algebra by means of formal vector fields. We show how this formula can be used to prove a convergent analogue of Guillemin and Sternberg’s Realization Theorem, which implies that the formal version and the convergent version of Lie’s conjecture are equivalent. We present sufficient conditions for our realization formula to yield only polynomial coefficients; they slightly generalize a known result. Next, we derive a sufficient, but very strong, condition for this formula to yield only coefficients in E. Finally, we prove Lie’s conjecture for n = 1, 2, and 3. In the first two cases, this result is not new, as Lie completely classified the transitive Lie algebras in those dimensions. For n = 3, the result could be considered new, as Lie did not publish his complete classification. Surprisingly, the sub-case that did not appear in print, is handled very easily by the strong condition mentioned before.

Modular Forms for GL(3) and Galois Representations

A description and an example are given of numerical experiments which look for a relation between modular forms for certain congruence subgroups of SL(3,ℤ) and Galois representations.

Ruled quartic surfaces, models and classification

New historical aspects of the classification, by Cayley and Cremona, of ruled quartic surfaces and the relation to string models and plaster models are presented. In a ‘modern’ treatment of the classification of ruled quartic surfaces the classical one is corrected and completed. The string models of Series XIII of some ruled quartic surfaces (manufactured by L. Brill and by M. Schilling) are based on a result of Rohn concerning curves in

Geometric aspects of the Painleve equations PIII(D 6 ) and PIII(D 7 )

{\mathbb{P}^1\times \mathbb{P}^1}

Hesse Pencils and 3-Torsion Structures

of bi-degree (2, 2). This is given here a conceptional proof.