A problem of complete intersections

Robbiano, Lorenzo

doi:10.1017/s0027763000015920

Cited by 7 publications

(10 citation statements)

References 6 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Let X = P m x P m ' and embed X in the protective space P n by: We do not know in general if the restriction about the characteristic of K to be zero is really necessary in corollary 4. However, theorem B and Robbiano's lemma above allow us to deduce the following result, which extends to arbitrary characteristic the result of Robbiano from [13] …”

Section: H\p O H P ) = H\p O H P ) = H\y 0%) -0 Pic (P) = Ff(pmentioning

confidence: 61%

A remark on the Grothendieck-Lefschetz theorem about the Picard group

Bădescu

1978

Nagoya Mathematical Journal

View full text Add to dashboard Cite

show abstract

Section: H\p O H P ) = H\p O H P ) = H\y 0%) -0 Pic (P) = Ff(pmentioning

confidence: 61%

A remark on the Grothendieck-Lefschetz theorem about the Picard group

Bădescu

1978

Nagoya Mathematical Journal

View full text Add to dashboard Cite

show abstract

“…Robbiano [28] proved this in the case where 5 is smooth and the ground field has characteristic zero. Since Γη Sing(S) = 0, S is normal.…”

Section: Lemma 132 Lei S C P 3 Be a Normal Surface Then Pic(s)/picmentioning

confidence: 87%

Applications of iterated curve blow-up to set theoretic complete intersections in 3.

Jaffe¹

1995

Journal Für Die Reine Und Angewandte Mathematik (Crelles Journal)

View full text Add to dashboard Cite

Introduction.We describe some new results on the set-theoretic complete intersection problem for projective space curves. Fix an algebraically closed ground field k. Let S, c= P 3 be surfaces. Suppose that Sr\ T is set-theoretically a smooth curve C of degree d and genusg. For purposes of the introduction, we label the main results äs A, B, Q, X, I, II, and III 2 ). The results I, II, and III are more technical than A, B, Q, and X.Suppose that S and T have no common singular points. We discover that this requirement imposes severe limitations. Indeed, theorem X asserts that if C is not a complete intersection, then degiS 1 ), deg(T) < 2d 4 . Fixing (d,g), one can in fact form a finite list of all possible pairs (deg(S), deg(T)), which is much shorter than the list implied by theorem X. For instance, when (d,g) -(4,0), and assuming for simplicity that deg(S) ^ deg(T), we find that (deg(S), deg(r)) e Very little is known about which of these degree pairs actually correspond to surface pairs (5, ).Suppose that S and T have only rational singularities, and that the ground field k has characteristic zero. We continue to assume that S and T have no common singular points. Under these conditions, we prove A that d^g + 3. (The actual Statement is somewhat stronger.)Suppose that S is normal, and that d > deg(S). Make no assumptions about how the singularities of S and T meet. Assume that char(fc) = 0. We show Q that C is linearly normal. In particular, it follows by Riemann-Roch that d & g + 3. *) Partially supported by the National Science Foundation.2 ) The actual numbering in the text is A = 12.2, B «11.11, Q = 11.8, X-13.1, 1-10.1, 11 = 10.3, 111 = 11.1. Brought to you by | University of Iowa Libraries Authenticated Download Date | 5/27/15 4:32 PM 2 Jaffe, Set theore c complete intersectionsSuppose that S is a quartic surface having only rational singularities. Allow T to be an arbitrary surface, and make no assumptions about how the singularities of S and T meet. Assume that char(fc) = 0. Under these conditions, we prove B that C is linearly normal.In other papers [17], [18], we have proved the following complementary results (in characteristic zero): if S has only ordinary nodes s singularities, or is a cone, or has degree ^ 3, then d^ g + 3. It is conceivable (in characteristic zero) that this inequality is valid without any restrictions whatsoever on S and Γ, or even that C is always linearly normal. Examples of smooth set-theoretic complete intersection curves in CP 3 have been constructed by Gallarati [8], Catanese [3], Rao ([27], prop. 14), and the author [19].To explain the results I, II, and III, and to describe the methods by which we prove A, B, and X, there are two key ideas which must be discussed 3 ). Both of these ideas have to do with the iterated blowing up of curves.The first idea has to do with certain invariants p { = p t (S 9 C) (/e N) which we associate to a pair (S, C) consisting of an abstract surface S and a smooth curve C on S such that CcJ: Sing (S). Let π : S -»S be the blow-up along C. The...

show abstract

“…It is known ( [6]) that a noncomplete intersection curve cannot be a set-theoretic complete intersection on a nonsingular surface. That is the reason why the singular surfaces come into play in this paper.…”

Section: Introductionmentioning

confidence: 99%

On the curves of contact on surfaces in a projective space. III

Boratyński

1997

Proc. Amer. Math. Soc.

View full text Add to dashboard Cite

Abstract. Suppose a smooth curve C is a set-theoretic complete intersection of two surfaces F and G with the multiplicity of F along C less than or equal to the multiplicity of G along C. One obtains a relation between the degrees of C, F and G, the genus of C, and the multiplicity of F along C in case F has only ordinary singularities. One obtains (in the characteristic zero case) that a nonsingular rational curve of degree 4 in P 3 is not settheoretically an intersection of 2 surfaces, provided one of them has at most ordinary singularities. The same result holds for a general nonsingular rational curve of degree ≥ 5.

show abstract

A problem of complete intersections

Cited by 7 publications

References 6 publications

A remark on the Grothendieck-Lefschetz theorem about the Picard group

A remark on the Grothendieck-Lefschetz theorem about the Picard group

Applications of iterated curve blow-up to set theoretic complete intersections in 3.

On the curves of contact on surfaces in a projective space. III

Contact Info

Product

Resources

About