Varieties Of Minimal Degree Research Articles

Cones between the cones of positive semidefinite forms and sums of squares

Abstract For n, d ∈ ℕ, the cone 𝓟 n+1,2d of positive semidefinite real forms in n + 1 variables of degree 2d contains the subcone Σ n+1,2d of those representable as finite sums of squares of real forms. Hilbert [11] proved that these cones coincide exactly in the Hilbert cases (n + 1, 2d) with n + 1 = 2 or 2d = 2 or (n + 1, 2d) = (3, 4). In this paper, we induce a filtration of intermediate cones between Σ n+1,2d and 𝓟 n+1,2d via the Gram matrix approach in [4] on a filtration of irreducible projective varieties V k−n ⊊ … ⊊ Vn ⊊ … ⊊ V 0 containing the Veronese variety. Here, k is the dimension of the vector space of real forms in n + 1 variables of degree d. By showing that V 0, …, V n (and V n+1 when n = 2) are varieties of minimal degree, we demonstrate that the corresponding intermediate cones coincide with Σ n+1,2d . We moreover prove that, in the non-Hilbert cases of (n + 1)-ary quartics for n ≥ 3 and (n + 1)-ary sextics for n ≥ 2, all the remaining cone inclusions are strict.

Characterization of projective varieties beyond varieties of minimal degree and del Pezzo varieties

Varieties of minimal degree and del Pezzo varieties are basic objects in projective algebraic geometry. Those varieties have been characterized and classified for a long time in many aspects. Motivated by the question “which varieties are the most basic and simplest except the above two kinds of varieties in view of geometry and syzygies?”, we give an upper bound of the graded Betti numbers in the quadratic strand and characterize the extremal cases.The extremal varieties of dimension n, codimension e, and degree d are exactly characterized by the following two types:(i)Varieties with d=e+2, depthX=n, and Green-Lazarsfeld index a(X)=0,(ii)Arithmetically Cohen-Macaulay varieties with d=e+3. This is a generalization of G. Castelnuovo, G. Fano, and E. Park's results on the number of quadrics and an extension of the characterizations of varieties of minimal degree and del Pezzo varieties in view of linear syzygies of quadrics due to K. Han and S. Kwak ([6,8,30,16]).In addition, we show that every variety X that belongs to (i) or (ii) is always contained in a unique rational normal scroll Y as a divisor. Also, we describe the divisor class of X in Y.

K3 carpets on minimal rational surfaces and their smoothings

In this paper, we study K3 double structures on minimal rational surfaces [Formula: see text]. The results show there are infinitely many non-split abstract K3 double structures on [Formula: see text] parametrized by [Formula: see text], countably many of which are projective. For [Formula: see text] there exists a unique non-split abstract K3 double structure which is non-projective (see [J.-M. Drézet, Primitive multiple schemes, preprint (2020), arXiv:2004.04921, to appear in Eur. J. Math.]). We show that all projective K3 carpets can be smoothed to a smooth K3 surface. One of the byproducts of the proof shows that unless [Formula: see text] is embedded as a variety of minimal degree, there are infinitely many embedded K3 carpet structures on [Formula: see text]. Moreover, we show any embedded projective K3 carpet on [Formula: see text] with [Formula: see text] arises as a flat limit of embeddings degenerating to 2:1 morphism. The rest do not, but we still prove the smoothing result. We further show that the Hilbert points corresponding to the projective K3 carpets supported on [Formula: see text], embedded by a complete linear series are smooth points if and only if [Formula: see text]. In contrast, Hilbert points corresponding to projective (split) K3 carpets supported on [Formula: see text] and embedded by a complete linear series are always smooth. The results in [P. Bangere, F. J. Gallego and M. González, Deformations of hyperelliptic and generalized hyperelliptic polarized varieties, preprint (2020), arXiv:2005.00342] show that there are no higher dimensional analogues of the results in this paper.

Ramification Divisors of General Projections

We study ramification divisors of projections of a smooth projective variety onto a linear space of the same dimension. We prove that for a large class of varieties, the ramification divisors of such projections vary in a maximal dimensional family. We study the map that associates to a linear projection its ramification divisor. By a degeneration argument involving (linked) limit linear series of higher rank, we show that this map is dominant for most (but not all!) varieties of minimal degree.

On 3-linear varieties of codimension 2

A projective variety in a projective space is said to be [Formula: see text]-linear if it is [Formula: see text]-regular and has no defining equation of degree [Formula: see text]. It is well known that [Formula: see text]-linear varieties are exactly varieties of minimal degree. In this paper, we study [Formula: see text]-linear varieties of codimension [Formula: see text]. We classify all smooth [Formula: see text]-linear varieties of codimension 2. There are six kinds of such varieties. Also, we provide some nonconic singular [Formula: see text]-linear varieties of codimension [Formula: see text].

Low-Rank Sum-of-Squares Representations on Varieties of Minimal Degree

A celebrated result by Hilbert says that every real nonnegative ternary quartic is a sum of three squares. We show more generally that every nonnegative quadratic form on a real projective variety $X$ of minimal degree is a sum of $\dim(X)+1$ squares of linear forms. This strengthens one direction of a recent result due to Blekherman, Smith, and Velasco. Our upper bound is the best possible, and it implies the existence of low-rank factorizations of positive semidefinite bivariate matrix polynomials and representations of biforms as sums of few squares. We determine the number of equivalence classes of sum-of-squares representations of general quadratic forms on surfaces of minimal degree, generalizing the count for ternary quartics by Powers, Reznick, Scheiderer, and Sottile.

Interpolation of Varieties of Minimal Degree

It is well known that one can find a rational normal curve in $\mathbb P^n$ through $n+3$ general points. We prove a generalization of this to higher dimensional varieties, showing that smooth varieties of minimal degree can be interpolated through points and linear spaces.

Deformations of canonical triple covers

In this paper, we show that if X is a smooth variety of general type of dimension m≥3 for which the canonical map induces a triple cover onto Y, where Y is a projective bundle over P1 or onto a projective space or onto a quadric hypersurface, embedded by a complete linear series (except Q3 embedded in P4), then the general deformation of the canonical morphism of X is again canonical and induces a triple cover. The extremal case when Y is embedded as a variety of minimal degree is of interest, due to its appearance in numerous situations. For instance, by looking at threefolds Y of minimal degree we find components of the moduli of threefolds X of general type with KX3=3pg−9,KX3≠6, whose general members correspond to canonical triple covers. Our results are especially interesting as well because they have no lower dimensional analogues.

On non-normal del Pezzo varieties

Two projective subvarieties of Pr are said to be projectively equivalent if they are identified by a coordinate change. Up to projective equivalence, varieties of minimal degree were completely classified more than one hundred years ago by P. del Pezzo and E. Bertini.As the next case, we study the same problem for del Pezzo varieties, focusing on the non-normal case of degree ⩾5. Note that the cases of degrees 3 and 4 were dealt with in Lee et al. (2011) [8] and Lee et al. (2012) [9], respectively. Our main result, Theorem 4.1, provides a complete classification of non-normal del Pezzo varieties of degree at least 5, up to projective equivalence.

Buchsbaum varieties with next to sharp bounds on Castelnuovo-Mumford regularity

This paper is devoted to the study of the next extremal case for a Castelnuovo-type bound regV ≤ ?(deg V―1)/codim V?+1 of the Castelnuovo-Mumford regularity for a nondegenerate Buchsbaum variety V. A Buchsbaum variety with the maximal regularity is known to be a divisor on a variety of minimal degree if the degree of the variety is large enough. We show that a Buchsbaum variety satisfying regV = ?(deg V — 1)/codim V? is a divisor on a Del Pezzo variety if deg V ≫ 0.

On varieties of almost minimal degree I: Secant loci of rational normal scrolls

To provide a geometrical description of the classification theory and the structure theory of varieties of almost minimal degree, that is of non-degenerate irreducible projective varieties whose degree exceeds the codimension by precisely 2, a natural approach is to investigate simple projections of varieties of minimal degree. Let X ̃ ⊂ P K r + 1 be a variety of minimal degree and of codimension at least 2, and consider X p = π p ( X ̃ ) ⊂ P K r where p ∈ P K r + 1 ∖ X ̃ . By Brodmann and Schenzel (2007) [1], it turns out that the cohomological and local properties of X p are governed by the secant locus Σ p ( X ̃ ) of X ̃ with respect to p . Along these lines, the present paper is devoted to giving a geometric description of the secant stratification of X ̃ , that is of the decomposition of P K r + 1 via the types of secant loci. We show that there are at most six possibilities for the secant locus Σ p ( X ̃ ) , and we precisely describe each stratum of the secant stratification of X ̃ , each of which turns out to be a quasi-projective variety. As an application, we obtain a different geometrical description of non-normal del Pezzo varieties X ⊂ P K r , first classified by Fujita (1985) [3, Theorem 2.1(a)] by providing a complete list of pairs ( X ̃ , p ) , where X ̃ ⊂ P K r + 1 is a variety of minimal degree, p ∈ P K r + 1 ∖ X ̃ and X p = X ⊂ P K r .

Germs of integrable forms and varieties of minimal degree

We study the subvariety of integrable 1-forms in a finite-dimensional vector space W ⊂ Ω 1 ( C n , 0 ) . We prove that the irreducible components with dimension comparable with the rank of W are of minimal degree.

Small schemes and varieties of minimal degree

We prove that if X ⊂ P r is any 2-regular scheme (in the sense of Castelnuovo-Mumford) then X is small . This means that if L is a linear space and Y := L ∩ X is finite, then Y is linearly independent in the sense that the dimension of the linear span of Y is deg Y + 1. The converse is true and well-known for finite schemes, but false in general. The main result of this paper is that the converse, "small implies 2-regular", is also true for reduced schemes (algebraic sets). This is proven by means of a delicate geometric analysis, leading to a complete classification: we show that the components of a small algebraic set are varieties of minimal degree, meeting in a particularly simple way. From the classification one can show that if X ⊂ P r is 2-regular, then so is X red , and so also is the projection of X from any point of X . Our results extend the Del Pezzo-Bertini classification of varieties of minimal degree, the characterization of these as the varieties of regularity 2 by Eisenbud-Goto, and the construction of 2-regular square-free monomial ideals by Fröberg.

On Singular Varieties Having an Extremal Secant Line

ABSTRACT Let X be a nondegenerate subvariety of degree d and codimension e in the projective space ℙ n . If X is smooth, any multisecant line to X cuts X along a 0-dimensional scheme of length at most d − e + 1. Moreover, smooth varieties X having a (d − e + 1)-secant line (an extremal secant line) have been completely classified, extending del Pezzo and Bertini classification of varieties of minimal degree. In this article, we almost completely classify possibly singular varieties having an extremal secant line, without any assumptions on the singularities of X. First, we show that, if e ≠ 2, a multisecant line to X meets X along a 0-dimensional scheme of length at most d − e + 1. Then, we completely classify singular varieties having a (d − e + 1)-secant line for e ≠ 3. A partial result is provided in case e = 3.

Separators of points on algebraic surfaces

For a finite set of points X ⊆ P n and for a given point P ∈ X , the notion of a separator of P in X (a hypersurface containing all the points in X except P ) and of the degree of P in X , d X P (the minimum degree of these separators) has been largely studied. In this paper we extend these notions to a set of points X on a projectively normal surface S ⊆ P n , considering as separators arithmetically Cohen–Macaulay curves and generalizing the case S = P 2 in a natural way. We denote the minimum degree of such curves as d X , S P and we study its relation to d X P . We prove that if S is a variety of minimal degree these two terms are explicitly related by a formula, whereas only an inequality holds for other kinds of surfaces.

Certain Minimal Varieties Are Set-Theoretic Complete Intersections

We present a class of homogeneous ideals which are generated by monomials and binomials of degree 2 and are set-theoretic complete intersections. This class includes certain reducible varieties of minimal degree and, in particular, the presentation ideals of the fiber cone algebras of monomial varieties of codimension 2.

Arithmetically Buchsbaum divisors on varieties of minimal degree

Arithmetically Buchsbaum divisors on varieties of minimal degree

Multiplicity andt-isomultiple ideals

LetVbe an irreducible non degenerate variety inPn;a classical geometric result says that degree (V) ≥ codimV+1and, if equality holds,Vis said to be of minimal degree. Varieties of minimal degree has been classified by Del Pezzo and Bertini and they all are intersections of quadrics. The local version of this result is due to J. Sally who proved that ifis a regular local ring andis a Cohen-Macaulay local ring of minimal multiplicity, according to the bounde(R) ≥ height (I) + 1 given by Abhyankar, then the tangent coneofRis intersection of quadrics and it is Cohen-Macaulay.