Hyperplane Sections Research Articles

Characteristic conic connections and torsion-free principal connections

We study the relation between torsion tensors of principal connections on G-structures and characteristic conic connections on associated cone structures. We formulate sufficient conditions under which the existence of a characteristic conic connection implies the existence of a torsion-free principal connection. We verify these conditions for adjoint varieties of simple Lie algebras, excluding those of type Aℓ≠2 or Cℓ. As an application, we give a complete classification of the germs of minimal rational curves whose VMRT at a general point is such an adjoint variety: nontrivial ones come from lines on hyperplane sections of certain Grassmannians or minimal rational curves on wonderful group compactifications.

Derived equivalence and homological projective duality in GLSM

Brane transport provides a way to realize functors between the categories of B-branes of different phases in gauged linear sigma model (GLSM). When appropriately designed, these functors induce derived equivalence between Calabi–Yau manifolds. In some cases, brane transport can also be used to extract the homological projective dual (HPD) of certain projective embeddings. We describe the GLSMs realizing derived equivalence between different geometries via brane transport with emphasis on the nonabelian models, and how to construct the GLSMs where brane transport gives rise to the embedding of HPD category into the derived category of universal hyperplane section. The latter gives rise to an explanation of the relationship between GLSM and HPD, and also a noncommutative geometric interpretation to HPD.

Curves on Brill–Noether special K3 surfaces

AbstractMukai showed that projective models of Brill–Noether general polarized K3 surfaces of genus 6–10 and 12 are obtained as linear sections of projective homogeneous varieties, and that their hyperplane sections are Brill–Noether general curves. In general, the question, raised by Knutsen, and attributed to Mukai, of whether the Brill–Noether generality of any polarized K3 surface is equivalent to the Brill–Noether generality of smooth curves in the linear system , is still open. Using Lazarsfeld–Mukai bundle techniques, we answer this question in the affirmative for polarized K3 surfaces of genus , which provides a new and unified proof even in the genera where Mukai models exist.

Ample cones of Hilbert schemes of points on hypersurfaces in ℙ³

Let X X be a very general degree d ≥ 5 d\geq 5 hypersurface in P 3 \mathbb {P}^3 . We compute the ample cone of the Hilbert scheme X [ n ] X^{[n]} of n n points on X X for various small values of n n (the answer is already known for large n n ). We obtain complete answers in some cases and find lower bounds in certain others. We also observe that in the case of X [ 2 ] X^{[2]} for quintic hypersurfaces X X , the existence (or absence) of hyperplane sections with points of high multiplicity also plays a role in the answer to the question at hand, in contrast with cases known earlier. Finally, in the case that a degree d ≥ 3 d\geq 3 smooth hypersurface X X contains a line, we compute the nef cone of X [ n ] X^{[n]} in a slice of the Néron-Severi space.

Extensions of curves with high degree with respect to the genus

We classify linearly normal surfaces $S \subset \mathbf{P}^{r+1}$ of degree $d$ such that $4g-4 \leq d \leq 4g+4$, where $g>1$ is the sectional genus (it is a classical result that for larger $d$ there are only cones). We apply this to the study of the extension theory of pluricanonical curves and genus $3$ curves, whenever they verify Property $N_2$, using and slightly expanding the theory of integration of ribbons of the authors and E.~Sernesi. We compute the corank of the relevant Gaussian maps, and we show that all ribbons over such curves are integrable, and thus there exists a universal extension. We carry out a similar program for linearly normal hyperelliptic curves of degree $d\geq 2g+3$. We classify surfaces having such a curve $C$ as a hyperplane section, compute the corank of the relevant Gaussian maps, and prove that all ribbons over $C$ are integrable if and only if $d=2g+3$. In the latter case we obtain the existence of a universal extension.

Irreducibility of moduli of vector bundles over a very general sextic surface

Let S be a very general smooth hypersurface of degree 6 in P3. In this paper we will prove that the moduli space of μ-stable rank 2 torsion free sheaves with respect to hyperplane section having c1=OS(1), with fixed, c2≥27 is irreducible.

Hyperplane sections of non-standard graded rings

The purpose of this paper is to explain a method on the generalization of the Bertini-type theorem on standard graded rings to the non-standard graded case of certain type.

Rabinowitz Floer homology for prequantization bundles and Floer Gysin sequence

Let Y be a prequantization bundle over a closed spherically monotone symplectic manifold Σ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Sigma $$\\end{document}. Adapting an idea due to Diogo and Lisi, we study a split version of Rabinowitz Floer homology for Y in the following two settings. First, Σ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Sigma $$\\end{document} is a symplectic hyperplane section of a closed symplectic manifold X satisfying a certain monotonicity condition; in this case, X\\Σ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$X {{\\setminus }} \\Sigma $$\\end{document} is a Liouville filling of Y. Second, the minimal Chern number of Σ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Sigma $$\\end{document} is greater than one, which is the case where the Rabinowitz Floer homology of the symplectization R×Y\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {R} \ imes Y$$\\end{document} is defined. In both cases, we construct a Gysin-type exact sequence connecting the Rabinowitz Floer homology of X\\Σ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$X{\\setminus }\\Sigma $$\\end{document} or R×Y\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {R} \ imes Y$$\\end{document} and the quantum homology of Σ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Sigma $$\\end{document}. As applications, we discuss the invertibility of a symplectic hyperplane section class in quantum homology, the isotopy problem for fibered Dehn twists, the orderability problem for prequantization bundles, and the existence of translated points. We also provide computational results based on the exact sequence that we construct.

Effective homology and periods of complex projective hypersurfaces

We introduce a new algorithm for computing the periods of a smooth complex projective hypersurface. The algorithm intertwines with a new method for computing an explicit basis of the singular homology of the hypersurface. It is based on Picard–Lefschetz theory and relies on the computation of the monodromy action induced by a one-parameter family of hyperplane sections on the homology of a given section. We provide a SageMath implementation. For example, on a laptop, it makes it possible to compute the periods of a smooth complex quartic surface with hundreds of digits of precision in typically an hour.

Surfaces with Prym-canonical hyperplane sections

In this paper, we will describe some general properties regarding surfaces with Prym-canonical hyperplane sections, determining also important conditions on the geometric genera of the possible singularities that such a surface can have. Moreover, we will construct new examples of this type of surfaces.

Transverse linear subspaces to hypersurfaces over finite fields

Ballico proved that a smooth projective variety X of degree d and dimension m over a finite field of q elements admits a smooth hyperplane section if q≥d(d−1)m. In this paper, we refine this criterion for higher codimensional linear sections on smooth hypersurfaces and for hyperplane sections on Frobenius classical hypersurfaces. We also prove a similar result for the existence of reduced hyperplane sections on reduced hypersurfaces.

Remarks on the geometry of the variety of planes of a cubic fivefold

This note presents some properties of the variety of planes $F_2(X)\subset G(3,7)$ of a cubic $5$-fold $X\subset \mathbb P^6$. A cotangent bundle exact sequence is first derived from the remark made by Iliev and Manivel that $F_2(X)$ sits as a Lagrangian subvariety of the variety of lines of a cubic $4$-fold, which is a hyperplane section of $X$. Using the sequence, the Gauss map of $F_2(X)$ is then proven to be an embedding. The last section is devoted to the relation between the variety of osculating planes of a cubic $4$-fold and the variety of planes of the associated cyclic cubic $5$-fold.

Classification of subpencils for hyperplane sections on certain K3 surfaces

AbstractWe classify the subpencils of complete linear systems for the hyperplane sections on K3 surfaces obtained as the complete intersection of a hyperquadric and a hypercubic. The classification is done from three points of view, namely, the type of a general fibre, the base locus and the Horikawa index of the essential member. This classification shows the distinct phenomenons depending on the rank of the hyperquadrics containing the surface.

On the topology of the transversal slice of a quasi-homogeneous map germ

AbstractWe consider a corank 1, finitely determined, quasi-homogeneous map germ f from $(\mathbb{C}^2,0)$ to $(\mathbb{C}^3,0)$ . We describe the embedded topological type of a generic hyperplane section of $f(\mathbb{C}^2)$ , denoted by $\gamma_f$ , in terms of the weights and degrees of f. As a consequence, a necessary condition for a corank 1 finitely determined map germ $g\,{:}\,(\mathbb{C}^2,0)\rightarrow (\mathbb{C}^3,0)$ to be quasi-homogeneous is that the plane curve $\gamma_g$ has either two or three characteristic exponents. As an application of our main result, we also show that any one-parameter unfolding $F=(f_t,t)$ of f which adds only terms of the same degrees as the degrees of f is Whitney equisingular.

On the weak Lefschetz property for height four equigenerated complete intersections

We consider the conjecture that all artinian height 4 complete intersections of forms of the same degree d d have the Weak Lefschetz Property (WLP). We translate this problem to one of studying the general hyperplane section of a certain smooth curve in P 3 \mathbb P^3 , and our main tools are the Socle Lemma of Huneke and Ulrich together with a careful liaison argument. Our main results are (i) a proof that the property holds for d = 3 , 4 d=3,4 and 5; (ii) a partial result showing maximal rank in a non-trivial but incomplete range, cutting in half the previous unknown range; and (iii) a proof that maximal rank holds in a different range, even without assuming that all the generators have the same degree. We furthermore conjecture that if there were to exist any height 4 complete intersection generated by forms of the same degree and failing WLP then there must exist one (not necessarily the same one) failing by exactly one (in a sense that we make precise). Based on this conjecture we outline an approach to proving WLP for all equigenerated complete intersections in four variables. Finally, we apply our results to the Jacobian ideal of a smooth surface in P 3 \mathbb P^3 .

Self-Dual Matroids from Canonical Curves

Self-dual configurations of 2n points in a projective space of dimension n – 1 were studied by Coble, Dolgachev–Ortland, and Eisenbud–Popescu. We examine the self-dual matroids and self-dual valuated matroids defined by such configurations, with a focus on those arising from hyperplane sections of canonical curves. These objects are parametrized by the self-dual Grassmannian and its tropicalization. We tabulate all self-dual matroids up to rank 5 and investigate their realization spaces. Following Bath, Mukai, and Petrakiev, we explore algorithms for recovering a curve from the configuration. A detailed analysis is given for self-dual matroids arising from graph curves.

Gauss‐Prym maps on Enriques surfaces

AbstractWe prove that the kth Gaussian map is surjective on a polarized unnodal Enriques surface with . In particular, as a consequence, when , we obtain the surjectivity of the kth Gauss‐Prym map , with , on smooth hyperplane sections . In case , it is sufficient to ask .

On star-convex bodies with rotationally invariant sections

We will prove that an origin-symmetric star-convex body K with sufficiently smooth boundary and such that every hyperplane section of K passing through the origin is a body of affine revolution, is itself a body of affine revolution. This will give a positive answer to the recent question asked by Bor, Hernández-Lamoneda, Jiménez de Santiago, and Montejano-Peimbert, though with slightly different prerequisites.

Rational ruled surfaces as symplectic hyperplane sections

We study embeddability of rational ruled surfaces as symplectic hyperplane sections into closed integral symplectic manifolds. From this we obtain results on Stein fillability of Boothby–Wang bundles over rational ruled surfaces.

Complex Links and Hilbert–Samuel Multiplicities

We describe a framework for estimating Hilbert–Samuel multiplicities for pairs of projective varieties from finite point samples rather than defining equations. The first step involves proving that this multiplicity remains invariant under certain hyperplane sections which reduce to a point and to a curve . Next, we establish that equals the Euler characteristic (and hence the cardinality) of the complex link of in . Finally, we provide explicit bounds on the number of uniform point samples needed (in an annular neighborhood of in ) to determine this Euler characteristic with high confidence.