Isometric embeddings of real projective spaces into Euclidean spaces

This paper studies isometric embeddings of RPn via non-degenerate symmetric bilinear maps. The main result shows the infimum dimension of target Euclidean spaces among these constructions for RPn is . Next, we construct Veronese maps by induction, which realize the infimum. Finally, we give a simple proof of Rigidity Theorem of Veronese maps.     

Some new embeddings and nonimmersions of real projective spaces

We obtain new families of embeddings and nonimmersions of real projective spaces in Euclidean space. The method involves obstruction theory, and includes several new insights.     

On embeddings of modular curves in projective spaces

We use explicit results on modular forms (Mui?, Ramanujan J 27:188–208, 2012) via uniformization theory to obtain embeddings of modular curves and more generally of compact Riemann surfaces attached to Fuchsian groups of the first kind in certain projective spaces. We obtain families of embeddings which vary smoothly with respect to a parameter in the upper-half plane. We study local expression for the divisors attached to the maps in the family.     

Non-special divisors on real algebraic curves and embeddings into real projective spaces

We show that there is a large class of non-special divisors of relatively small degree on a given real algebraic curve. If the real algebraic curve has many real components, such a divisor gives rise to an embedding (birational embedding, resp.) of the real algebraic curve into the real projective space ℙ ^r for r≥3 (r=2, resp.). We study these embeddings in quite some detail. Received: October 17, 2001?Published online: February 20, 2003     

Generalized Veronesean embeddings of projective spaces

We classify all embeddings θ: PG(n, q) → PG(d, q), with $d geqslant tfrac{{n(n + 3)}} {2}$d geqslant tfrac{{n(n + 3)}} {2}, such that θ maps the set of points of each line to a set of coplanar points and such that the image of θ generates PG(d, q). It turns out that d = ?n(n+3) and all examples are related to the quadric Veronesean of PG(n, q) in PG(d, q) and its projections from subspaces of PG(d, q) generated by sub-Veroneseans (the point sets corresponding to subspaces of PG(n, q)). With an additional condition we generalize this result to the infinite case as well.     

On the weak non-defectivity of veronese embeddings of projective spaces

Fix integers n, x, k such that n≥3, k>0, x≥4, (n, x)≠(3, 4) and k(n+1)<( _n ^n+x ). Here we prove that the order x Veronese embedding ofP ⁿ is not weakly (k−1)-defective, i.e. for a general S⊃P ⁿ such that #(S) = k+1 the projective space | I _2S (x)| of all degree t hypersurfaces ofP ⁿ singular at each point of S has dimension ( _n ^/n+x )−1− k(n+1) (proved by Alexander and Hirschowitz) and a general F∈| I _2S (x)| has an ordinary double point at each P∈ S and Sing (F)=S. The author was partially supported by MIUR and GNSAGA of INdAM (Italy).     

On the genus of real projective spaces

We construct a crystallization of the real projective space whose associated contracted complex is minimal with respect to the number of n-simplexes. Then we compute the regular genus of , which is the minimum genus of a closed connected surface into which a crystallization of regularly embeds. Received: 7 February 2007     

Hyperplanes and projective embeddings of dual polar spaces

This paper is based on the invited talk of the author at the Combinatorics 2010 conference which was held in Verbania (Italy) from June 27th till July 3rd 2010. It discusses hyperplanes and full projective embeddings of dual polar spaces, as well as some of their mutual connections. Many of the discussed results are very recent.     

Codimension 2 and 3 pluricanonical embeddings in projective spaces

In this paper some non existence results are given for smooth varieties of dimension bigger then 1, embedded in projective spaces with low codimension, by the pluricanonical system |mK _X |, withm≥2. The only meaningful cases of codimension 2 are surfaces in P⁴ and threefolds in P⁵, whose existence is excluded. When the codimension is 3, for surfaces in P⁵ is proven thatm=2, while for three-folds in P⁶ and fourfolds in P⁷ only two numerical possibilities for the degree are given.     

On projective invariants of the complex Finsler spaces

In this paper we extend the results on projective changes of complex Finsler metrics obtained in Aldea and Munteanu (2012) [3], by the study of projective curvature invariants of a complex Finsler space. By means of these invariants, the notion of complex Douglas space is then defined. A special approach is devoted to the obtaining of equivalence conditions for a complex Finsler space to be a Douglas one. It is shown that any weakly Kähler Douglas space is a complex Berwald space. A projective curvature invariant of Weyl type characterizes complex Berwald spaces. These must be either purely Hermitian of constant holomorphic curvature, or non-purely Hermitian of vanishing holomorphic curvature. Locally projectively flat complex Finsler metrics are also studied.     

On K-spanned embeddings of projective manifolds

We show that h⁰(X, L)≥2k for every k-spanned line bundle on a smooth complete complex surface X.     

Resolutions of Segre embeddings of projective spaces of any dimension

This paper deals with syzygies of the ideals of Segre embeddings. Let d≥3 and . We prove that satisfies Green-Lazarsfeld’s property N_p if and only if p≤3.     

Some remarks on embeddings of the flag geometries of projective planes in finite projective spaces

The flag geometry of a finite projective plane II of orders is the generalized hexagon of order (s, 1) obtained from II by putting equal to the set of all flags of II, by putting equal to the set of all points and lines of II and where I is the natural incidence relation (inverse containment), that is, is the dual of the double of II in the sense of [8]. Then we say that is fully (and weakly) embedded in the finite projective space PG(d, q) if is a subgeometry of the natural point-line geometry associated with PG(d, q), if s=q, if the set of points of generates PG(d, q) (and if the set of points of not opposite any given point of does not generate PG(d, q)). We have classified all such embeddings in [3, 4, 5, 6]. In the present paper, we weaken the hypotheses in some special cases, and we give an alternative formulation of the classification.     

On projective embeddings of generic rational surfaces

Let R be a commutative ring and let ?(σ) be a Gabriel filter of R such that R is σ-noetherian. We discuss the decomposition of the σ ¹-torsion submodule of a σ-torsionfree R-module and characterize the σ ^l-injectivity of σ-closed R-modules through the σ _m-injectivity of modules over noetherian local rings (S, m). As an application, we obtain new criteria to determine injectivity of modules over noetherian rings, of finite Krull dimension, and Krull domains.     

Cohomology of configuration spaces of complex projective spaces

In this paper we compute topological invariants for some configuration spaces of complex projective spaces. We shall describe Sullivan models for these configuration spaces.