Johannes Huisman

The group of automorphisms of a real rational surface is n-transitive

Let X be a rational nonsingular compact connected real algebraic surface. Denote by Aut(X) the group of real algebraic automorphisms of X. We show that the group Aut(X) acts n-transitively on X, for all natural integers n. As an application we give a new and simpler proof of the fact that two rational nonsingular compact connected real algebraic surfaces are isomorphic if and only if they are homeomorphic as topological surfaces.

Real Quotient Singularities and Nonsingular Real Algebraic Curves in the Boundary of the Moduli Space

The quotient of a real analytic manifold by a properly discontinuous group action is, in general, only a semianalytic variety. We study the boundary of such a quotient, i.e., the set of points at which the quotient is not analytic. We apply the results to the moduli space Mg/∝ of nonsingular real algebraic curves of genus g (g≤2). This moduli space has a natural structure of a semianalytic variety. We determine the dimension of the boundary of any connected component of Mg/∝. It turns out that every connected component has a nonempty boundary. In particular, no connected component of Mg/∝ is real analytic. We conclude that Mg/∝ is not a real analytic variety.

Clifford's inequality for real algebraic curves

Abstract We improve Cliffords Inequality for real algebraic curves. As an application we improve Harnacks Inequality for real space curves having a certain number of pseudo-lines. Another application involves the number of ovals that a real space curve can have.

On the neutral component of the Jacobian of a real algebraic curve having many components

Abstract Let C be a real algebraic curve of genus g ≥ 1 having at least g real components. We show that there is an embedding of C into 2g as a curve of degree 3g which induces a group structure on a connected component X of the set of effective divisors on C of degree g. Moreover, after having chosen a base point O ϵ X, there is a natural isomorphism of X onto the neutral real component of the Jacobian of C. This furnishes an explicit description of the group structure on the neutral real component of the Jacobian of a real algebraic curve of genus g ≥ 1 having many real components. If g = 1, one recovers the geometric description of the group structure on the neutral real component of a real elliptic curve.

THE EXPONENTIAL SEQUENCE IN REAL ALGEBRAIC GEOMETRY AND HARNACK'S INEQUALITY FOR PROPER REDUCED REAL SCHEMES

ABSTRACT We introduce the analytification of a scheme locally of finite type over . On one has an exponential sequence if is reduced. This exponential sequence gives rise to an analytic description of the Picard group Pic( ) if is proper and reduced. Using this description we generalize Harnacks Inequality for real algebraic curves to arbitrary proper reduced real schemes.

Schottky Uniformization of Real Algebraic Curves and an Application to Moduli

We show that a nonsingular compact connected real algebraic curve can be uniformized by a real Schottky group, i. e., a Schottky group in PGL2(ℂ) which is actually contained in PGL2(ℝ). As an application we show that the set Mrpg/ℝ of isomorphismclasses of nonsingularcompact connected real algebraic curves of genus g having real points, has a structure of a semianalytic variety. We show that this structure coincides with the semianalytic structure on Mrpg/ℝ defined via real Teichmuller spaces.

ALGEBRAIC MODULI OF REAL ELLIPTIC CURVES

We study algebraic moduli of real generalized elliptic curves. For this, one needs to study algebraic families of such curves. The most suitable class of parameter spaces seems to be the class of Nash manifolds. It turns out, however, that real generalized elliptic curves do not have fine Nash moduli. The somewhat more restricted moduli problem of so-called oriented real generalized elliptic curves does have fine Nash moduli. We prove this by explicitly constructing a universal family of oriented real generalized elliptic curves over a Nash manifold. It will follow that real generalized elliptic curves have coarse Nash moduli. In fact, the coarse moduli space is the Nash manifold P 1(R). As a consequence, every real generalized elliptic curve E has a real j-invariant j R (E) ∈ R ∪ {∞}. Let E and F be real generalized elliptic curves. Then j R (E) = j R (F) if and only if E and F are isomorphic as real curves. We also give an explicit formula for the real j-invariant of a real generalized elliptic curve defined by the Weierstrass equation y 2 = x 3 + ax + b.

Real algebraic differential forms on complex algebraic varieties

Abstract We show that every de Rham cohomology class on a nonsingular quasiprojective complex algebraic variety can be realized by a real algebraic differential form.

Principal bundles over a smooth real projective curve of genus zero

Let H 0 denote the kernel of the endomorphism, defined by z → (z/z - ) 2 , of the real algebraic group given by the Weil restriction of C*. Let X be a nondegenerate anisotropic conic in P 2 R . The principal C*-bundle over the complexification X C , defined by the ample generator of Pic(X C ), gives a principal H 0 -bundle F H0 over X through a reduction of structure group. Given any principal G-bundle E G over X, where G is any connected reductive linear algebraic group defined over R, we prove that there is a homomorphism ρ : H 0 → G such that E G is isomorphic to the principal G-bundle obtained by extending the structure group of F H0 using ρ. The tautological line bundle over the real projective line P 1 R , and the principal Z/2Z- bundle P 1 C → P 1 R , together give a principal G m × (Z/2Z)-bundle F on P 1 R . Given any principal G-bundle E G over P 1 R , where G is any connected reductive linear algebraic group defined over R, we prove that there is a homomorphism ρ : G m × (Z/2Z) → G such that E G is isomorphic to the principal G-bundle obtained by extending the structure group of F using ρ.

Modules over twisted group rings and vector bundles over the anisotropic real conic

Abstract We prove, in an elementary way, that a locally free sheaf of finite rank over the anisotropic real conic is the direct sum of indecomposable locally free sheaves of rank 1 or 2. Our proof is purely algebraic, and is based on a classification of graded ℂ[X, Y]-modules endowed with a certain action of the cyclic group ℤ/4ℤ.