Surface In P3 Research Articles

A note on smooth surfaces in P-4

We exhibit a simpler and quick proof of a theorem proved by G. Ellingsrud and Ch. Peskine, which states that there are only a finite number of components in the Hilbert scheme parametrizing smooth surfaces in P4 not of general type.

Rational surfaces in ℙ4 containing a plane curve

The families of smooth rational surfaces in ℙ4 have been classified in degree ⩽ 10. All known rational surfaces in ℙ4 can be represented as blow-ups of the plane P2. The fine classification of these surfaces consists of giving explicit open and closed conditions which determine the configurations of points corresponding to all surfaces in a given family. Using a restriction argument originally due independently to Alexander and Bauer we achieve the fine classification in two cases, namely non-special rational surfaces of degree 9 and special rational surfaces of degree 8. The first case completes the fine classification of all non-special rational surfaces. In the second case we obtain a description of the moduli space as the quotient of a rational variety by the symmetric group S5. We also discuss in how far this method can be used to study other rational surfaces in ℙ4.

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.

Hilbert polynomials of smooth surfaces in P4

Hilbert polynomials of smooth surfaces in P4

The border cases of the lifting theorem for surfaces in P4.

The border cases of the lifting theorem for surfaces in P4.

On the resolution of a curve lying on a smooth cubic surface in 𝑃³

Let C C be any reduced and irreducible curve lying on a smooth cubic surface in P 3 {\mathbb {P}^3} . In this paper we determine the graded Betti numbers of the ideal sheaf J C {\mathcal {J}_C} .

Smooth non-special surfaces of ℙ4

We consider smooth non-degenerate surfaces in ℙ4, and prove that there is a finite number of such surfaces which are: (a) sectionally non-special, i.e.h1(OC(1))=0, where C is a general hyperplane section of S; or (b) not of general type and non-special (i.e. h1(OC(1))=0.

Coordinate gap and bidualily

For a varietyX inPn, we define a numberb2, called coordinate gap number,and prove thatb2>2 only ifX is not reflexive. Then, for a smooth surface inP3, we obtain a concrete sufficient and necessary condition forb2>2, which enables us to discuss the biduality of surfaces inP3.

The structure of an even liaison class

We describe a structure called the Lazarsfeld-Rao property for even liaison classes in projective space. This property holds for many even liaison classes of curves in P 3 {{\mathbf {P}}^3} . We give a procedure for showing that an even liaison class in codimension 2 2 possesses this property, and we prove it for a family of even liaison classes in codimension 2 2 in any P n , n ⩾ 3 {{\mathbf {P}}^n},\;n \geqslant 3 . However, we conjecture that it in fact holds for every even liaison class in codimension 2 2 , so we want to give consequences for an even liaison class that possesses this property. The main element in describing this structure is the notion of a basic double link. The Lazarsfeld-Rao property says that there exist minimal elements of the even liaison class and that any element of the even liaison class can be deformed to a curve obtained by a sequence of basic double links beginning with any minimal element. We show that there is a unique standard type of sequence for any given element of the even liaison class. As a result, we can express the even liaison class as a disjoint union of irreducible nonempty families parameterized by certain finite sequences of integers. The standard numerical invariants of the elements of any family can be computed from the associated sequence of integers. We apply this to surfaces in P 4 {{\mathbf {P}}^4} . Our main tool for these results is a deformation technique related to liaison in codimension 2 2 . We also study Schwartau’s procedure of Liaison Addition in codimension 2 2 from the point of view of vector bundles. Using this, we give a different sort of structure for an even liaison class with the Lazarsfeld-Rao property.

The smooth surfaces in P4 without apparent triple points

The smooth surfaces in P4 without apparent triple points

On the monodromy group of everywhere tangent lines to the octic surface in 𝑃³

Let S 0 {S_0} be an octic surface in P 3 , G = G ( 1 , 3 ) {{\mathbf {P}}^3},G = G(1,3) = Grassmannian of lines in P 3 {{\mathbf {P}}^3} , and J = { ( x , l ) | x ∈ l ∩ S 0 } ⊂ S 0 × G {\mathbf {J}} = \{ (x,l)|x \in l \cap {S_0}\} \subset {S_0} \times G . Then dim ⁡ J = 5 \dim {\mathbf {J}} = 5 . Let L = { l | l is e v e r y w h e r e tangent to S 0 } − ⊂ G {\mathbf {L}} = {\{ l|l{\text { is }}everywhere{\text { tangent to }}{S_0}\} ^ - } \subset G . Let π 2 : S 0 × G → G {\pi _2}:{S_0} \times G \to G be the projection onto the second factor. We denote its restriction to J {\mathbf {J}} also by π 2 {\pi _2} . Then the locus of everywhere tangent lines is π 2 ( L ) {\pi _2}({\mathbf {L}}) . In this article we show that the monodromy group of these lines is the full symmetric group.

Hodge numbers of linked surfaces in ℙ4

Hodge numbers of linked surfaces in ℙ4

Examples of abelian surfaces in P4.

Examples of abelian surfaces in P4.

The Horrocks-Mumford bundle and the Ferrand construction

There is an analogue of the Ferrand construction for smooth surfaces in ℙ4: if the normal bundle of such a surface X has a suitable 1-subbundle, then a 2-vector bundle can be constructed, which has a section vanishing doubly on X. In this way the Horrocks-Mumford bundle is recovered from the quintic scroll.

Degenerations of 𝐾3 surfaces of degree 4

A generic K 3 K3 surface of degree 4 4 may be embedded as a nonsingular quartic surface in P 3 {{\mathbf {P}}_3} . Let f : X → Spec C [ [ t ] ] f:X \to \operatorname {Spec} \;{\mathbf {C}}[[t]] be a family of quartic surfaces such that the generic fiber is regular. Let Σ 0 {\Sigma _0} , Σ 2 0 {\Sigma _2^0} , Σ 4 {\Sigma _4} be respectively a nonsingular quadric in P 3 {{\mathbf {P}}_3} , a cone in P 3 {{\mathbf {P}}_3} over a nonsingular conic and a rational, ruled surface in P 9 {{\mathbf {P}}_9} which has a section with self intersection − 4 - 4 . We show that there exists a flat, projective morphism f ′ : X ′ → Spec C [ [ t ] ] f’:X’ \to {\text {Spec}}\;{\mathbf {C}}[[t]] and a map ρ : Spec C [ [ t ] ] → Spec C [ [ t ] ] \rho :{\text {Spec}}\:{\mathbf {C}}[[t]] \to {\text {Spec}}\:{\mathbf {C}}[[t]] such that (i) the generic fiber of f ′ f’ and the generic fiber of the pull-back of f f via ρ \rho are isomorphic, (ii) the fiber X 0 ′ {X’_0} of f ′ f’ over the closed point of Spec C [ [ t ] ] {\text {Spec}}\;{\mathbf {C}}[[t]] has only insignificant limit singularities and (iii) X 0 ′ {X’_0} is either a quadric surface or a double cover of Σ 0 {\Sigma _0} , Σ 2 0 {\Sigma _2^0} or Σ 4 {\Sigma _4} . The theorem is proved using the geometric invariant theory.