Grothendieck–Lefschetz theory, set-theoretic complete intersections and rational normal scrolls

On determinantal ideals and algebraic dependence

“…If X contains at least one nilpotent or Jordan block, then one of the entries of X is zero or can be annihilated by elementary row operations, so that ara I 2 (X) ≤ 2n − 4 by Corollary 3.2. If, on the other hand, X is a concatenation of scroll blocks, then some of them have at least two columns, so that the claim follows by [1,Theorem 2] or by [3,Theorem 4.2] (and, for n > 3, also from Theorem 3.1).…”

“…, q ′ mn−t 2 ). The inclusion ⊃ is clear, since each q ′ h differs from q h by a multiple of ∆ in R. For the inclusion ⊂ it suffices to prove that whenever, for some x ∈ K mn , all polynomials q ′ h vanish at x, then ∆ also vanishes at x: in this case the same is true for all polynomials q h , hence, by (1), for all t-minors of X, and the claim follows by Hilbert's Nullstellensatz. Suppose by contradiction that, under the given assumption, ∆ does not vanish at x.…”

Seshadri positive submanifolds of polarized manifolds

“…The examples of Segre embeddings discussed above are especially interesting. Indeed, although the rational normal scrolls F e ֒→ P e+3 are all set-theoretic (but not schemetheoretic) complete intersections in P e+3 (see [33], or also [6]), the product P 2 × P 1 (respectively P 3 × P 1 ) is not set-theoretic complete intersection in P 5 (respectively in P 7 ). In fact, P 2 × P 1 (respectively P 3 × P 1 ) is not even the zero locus of a section of an ample rank 2 vector bundle on P 5 (respectively of an ample rank 3 vector bundle on P 7 ), see [26] for P 2 × P 1 and [6, Corollary 4.5] for P 3 × P 1 .…”

“…On the other hand, by combining Proposition 3.8 and Corollary 5.3, we know that the rational normal scroll F e in P e+3 , with e = 1, 2, 3, 4, is O P e+3 (1)-ample. By a result of Verdi [33] (see also [6,Corollary 4.2]), F e is a set-theoretic complete intersection in P e+3 . This implies that P e+3 \ F e is covered by k − 2 = e + 1 affine open subsets of P e+3 , whence cd(P e+3 \ F e ) ≤ e = k − 3.…”

Binomial edge ideals and rational normal scrolls

“…It is well-known that S/I X is Cohen-Macaulay and has an S-linear resolution. We refer the reader to [5], [4], [1] for properties of the ideal of the rational normal scroll.…”