[Paper Review] On smooth surfaces in P4 containing a plane curve

Philippe Ellia, C. Folegatti|ArXiv.org|Oct 31, 2003

Algebraic Geometry and Number Theory5 references3 citations

TL;DR

This paper investigates smooth surfaces in projective 4-space containing a plane curve, focusing on their geometric constraints when lying on hypersurfaces with singular loci. It proves that for smooth surfaces with irregularity zero lying on a quartic hypersurface with two-dimensional singular locus, the degree is effectively bounded by 40, offering a key classification tool for surfaces not of general type.

ABSTRACT

We consider smooth surfaces $S \subset \Pq$ containing a plane curve $P$ and prove some general result concerning the linear system $|H-P|$. We then look at regular surfaces lying on hypersurfaces of degree $s$ having a plane of multiplicity $(s-2)$. This implies that $S$ contains a plane curve. We prove that the degree of such surfaces is bounded and for $s=4$ we compute an actual bound.

Motivation & Objective

• To classify smooth surfaces in P4 containing a plane curve by analyzing their linear systems and geometric constraints.
• To establish effective degree bounds for such surfaces when embedded in hypersurfaces with controlled singularities.
• To investigate the role of the irregularity condition q(S)=0 in bounding surface degrees.
• To explore the interplay between the base locus of |H−P| and the residual scheme R in the plane intersection.
• To determine whether degree bounds persist without the q(S)=0 assumption, suggesting broader applicability.

Proposed method

• Analyzes the linear system |H−P| on a smooth surface S⊂P4 containing a plane curve P, showing its base locus is zero-dimensional and contained in the plane Π.
• Uses Severi’s theorem and Bertini’s theorem to prove that P is reduced and the general element of |H−P| is smooth outside Π.
• Defines the residual scheme R of S∩Π with respect to P and proves that the base locus of |H−P| equals R via Chern class computations.
• Applies adjunction and intersection theory to compute deg(R) = d−2p+P², linking it to invariants of the surface and the plane curve.
• Employs linkage theory and the theory of arithmetically Cohen-Macaulay (a.C.M.) curves to bound the geometric genus and degree of hyperplane sections.
• Uses the parametrization of quadrics Q_H through the plane to analyze the bidegree of curves Y_H and derive degree bounds via genus comparison.

Experimental results

Research questions

• RQ1 What is the maximum possible degree of a smooth surface S⊂P4 with q(S)=0 that lies on a quartic hypersurface Σ with dim(Sing(Σ))=2?
• RQ2 How does the presence of a plane curve P⊂S affect the structure of the linear system |H−P| and its base locus?
• RQ3 Can the degree of smooth surfaces containing a plane curve be effectively bounded when lying on hypersurfaces with (s−2)-uple planes?
• RQ4 Is the assumption q(S)=0 strictly necessary for degree bounds, or can they be extended to surfaces with positive irregularity?
• RQ5 What geometric conditions ensure that a surface S⊂P4 containing a plane curve is arithmetically Cohen-Macaulay?

Key findings

• The degree of smooth surfaces S⊂P4 with q(S)=0 lying on a quartic hypersurface Σ with dim(Sing(Σ))=2 is bounded above by 40.
• The base locus of the linear system |H−P| is zero-dimensional and coincides with the residual scheme R of S∩Π with respect to P.
• For surfaces of degree d≥21, no plane curve of degree p can exist if the surface lies on a quartic hypersurface with 2D singular locus.
• When d−p is odd, the genus of the hyperplane section Y_H∪P is bounded by a quadratic expression in d and p, leading to the same degree bound d≤20.
• The proof relies on comparing the geometric genus of hyperplane sections with the genus bound π−1≤d²/8, leading to a quadratic inequality that restricts d.
• The authors conjecture that degree bounds persist even without the q(S)=0 assumption, based on a.C.M. and covering space arguments on the desingularization of the hypersurface.