On the Singularities of Surfaces Ruled by Conics

We classify the singularities of a surface ruled by conics: they are rational double points of type A _n or D _n . This is proved by showing that they arise from a precise series of blow-ups of a suitable surface geometrically ruled by conics. We determine also the family of such surfaces which are birational models of a given surface ruled by conics and obtained in a “minimal way” from it.     

On Rational Surfaces Ruled by Conics

Abstract

We study projective rational surfaces ruled by conics, describing their singularities and special fibres. In particular, if Sis smooth, we give a “canonical” procedure to determine a minimal model among the geometrically ruled surfaces birational to S.     

Minimal discrepancies of toric singularities

The main purpose of this paper is to prove that minimal discrepancies ofn-dimensional toric singularities can accumulate only from above and only to minimal discrepancies of toric singularities of dimension less thann. I also prove that some lower-dimensional minimal discrepancies do appear as such limit.     

Embedded resolutions of non necessarily normal affine toric varietiesRésolution plongée d'une variété torique non nécessairement normale

We give a method to construct a partial embedded resolution of a nonnecessarily normal affine toric variety Z^Γ equivariantly embedded in a normal affine toric variety Z_ρ. This partial resolution is an embedded normalization inside a normal toric ambient space and a resolution of singularities of the ambient space, which always exists, provides an embedded resolution. The advantage is that this partial resolution is completely determined by the embedding Z^Γ?Z_ρ. As a by-product, the construction of the normalization is made without an explicit computation of the saturation of the semigroup Γ of the toric variety (see [3]). This result is valid for a base field k algebraically closed of arbitrary characteristic. To cite this article: P.D. González Pérez, B. Teissier, C. R. Acad. Sci. Paris, Ser. I 334 (2002) 379–382.     

On the normal bundle of curves on nodal quartic surfaces

Let k be an algebraically closed field, char k = 0. Let C be an irreducible nonsingular curve such that 2C = S ? F, where S and F are two surfaces and all the singularities of F are rational double points (if any). We prove that C can never pass through rational singularities of types A _2n n∈N, E₆ and E₈. We give conditions for C to pass through rational singularities of types. A _2k+1 k∈Z⁺ D_n n≥4 and E₇, (0.8).     

Combinatorics of the Toric Hilbert Scheme

The toric Hilbert scheme is a parameter space for all ideals with the same multigraded Hilbert function as a given toric ideal. Unlike the classical Hilbert scheme, it is unknown whether toric Hilbert schemes are connected. We construct a graph on all the monomial ideals on the scheme, called the flip graph, and prove that the toric Hilbert scheme is connected if and only if the flip graph is connected. These graphs are used to exhibit curves in P ⁴ whose associated toric Hilbert schemes have arbitrary dimension. We show that the flip graph maps into the Baues graph of all triangulations of the point configuration defining the toric ideal. Inspired by the recent discovery of a disconnected Baues graph, we close with results that suggest the existence of a disconnected flip graph and hence a disconnected toric Hilbert scheme. Received May 15, 2000, and in revised form March 8, 2001. Online publication January 7, 2002.     

Graph theoretic techniques in algebraic geometry II: construction of singular complex surfaces of the rational cohomology type of CP2

Methods of graph theory are used to obtain rational projective surfaces with only rational double points as singularities and with rational cohomology rings isomorphic to that of the complex projective plane. Uniqueness results for such cohomologyCP ²'s and for rational and integral homologyCP ²'s are given in terms of the typesA _k,D _k, orE _k of singularities allowed by the construction. Supported in part by National Science Foundation grant no. MCS 77-03540.     

Multi-valued solutions and branch point singularities for nonlinear hyperbolic or elliptic systems

Multi-valued solutions are constructed for 2 × 2 first-order systems using a generalization of the hodograph transformation. The solution is found as a complex analytic function on a complex Riemann surface for which the branch points move as part of the solution. The branch point singularities are envelopes for the characteristics and thus move at the characteristic speeds. We perform an analysis of stability of these singularities with respect to perturbations of the initial data. The generic singularity types are folds, cusps, and nondegenerate umbilic points with non-zero 3-jet. An isolated singularity is generically a square root branch point corresponding to a fold. Two types of collisions between singularities are generic: At a “tangential” collision between two singularities moving at the same characteristic speed, a cube root branch point is formed, corresponding to a cusp. A “non-tangential” collision, between two square root branch points moving at different characteristic speeds, remains a square root branch point at the collision and corresponds to a nondegenerate umbilic point. These results are also valid for a diagonalizable n-th order system for which there are exactly two speeds. © 1993 John Wiley & Sons, Inc.     

A boundedness result for toric log Del Pezzo surfaces

In this paper we give an upper bound for the Picard number of the rational surfaces which resolve minimally the singularities of toric log Del Pezzo surfaces of given index ℓ. This upper bound turns out to be a quadratic polynomial in the variable ℓ. Received: 18 June 2008     

Welschinger invariants of toric Del Pezzo surfaces with nonstandard real structures

The Welschinger invariants of real rational algebraic surfaces are natural analogs of the Gromov-Witten invariants, and they estimate from below the number of real rational curves passing through prescribed configurations of points. We establish a tropical formula for the Welschinger invariants of four toric Del Pezzo surfaces equipped with a nonstandard real structure. Such a formula for real toric Del Pezzo surfaces with a standard real structure (i.e., naturally compatible with the toric structure) was established by Mikhalkin and the author. As a consequence we prove that for any real ample divisor D on a surface Σ under consideration, through any generic configuration of c ₁(Σ)D − 1 generic real points, there passes a real rational curve belonging to the linear system |D|. To Vladimir Igorevich Arnold on the occasion of his 70th birthday     

Rigidity of certain admissible pairs of rational homogeneous spaces of Picard number 1 which are not of the subdiagram type

Recently, Mok and Zhang (2019) introduced the notion of admissible pairs (X₀, X) of rational homogeneous spaces of Picard number 1 and proved rigidity of admissible pairs (X₀, X) of the subdiagram type whenever X₀ is nonlinear. It remains unsolved whether rigidity holds when (X₀, X) is an admissible pair NOT of the subdiagram type of nonlinear irreducible Hermitian symmetric spaces such that (X₀, X) is nondegenerate for substructures. In this article we provide sufficient conditions for confirming rigidity of such an admissible pair. In a nutshell our solution consists of an enhancement of the method of propagation of sub-VMRT (varieties of minimal rational tangents) structures along chains of minimal rational curves as is already implemented in the proof of the Thickening Lemma of Mok and Zhang (2019). There it was proven that, for a sub-VMRT structure \(\overline{\omega} : \mathscr{C}(S) \rightarrow S\) on a uniruled projective manifold \((X,\,{\cal K})\) equipped with a minimal rational component and satisfying certain conditions so that in particular S is “uniruled” by open subsets of certain minimal rational curves on X, for a “good” minimal rational curve ? emanating from a general point x ∈ S, there exists an immersed neighborhood N_? of ? which is in some sense “uniruled” by minimal rational curves. By means of the Algebraicity Theorem of Mok and Zhang (2019), S can be completed to a projective subvariety Z ? X. By the author’s solution of the Recognition Problem for irreducible Hermitian symmetric spaces of rank ? 2 (2008) and under Condition (F), which symbolizes the fitting of sub-VMRTs into VMRTs, we further prove that Z is the image under a holomorphic immersion of X₀ into X which induces an isomorphism on second homology groups. By studying ?*-actions we prove that Z can be deformed via a one-parameter family of automorphisms to converge to X₀ ? X. Under the additional hypothesis that all holomorphic sections in Γ(X₀, T_x∣x₀) lift to global holomorphic vector fields on X, we prove that the admissible pair (X₀, X) is rigid. As examples we check that (X₀, X) is rigid when X is the Grassmannian G(n, n) of n-dimensional complex vector subspaces of W ? ?²ⁿ, n ? 3, and when X₀ ? X is the La grangian Grassmannian consisting of Lagrangian vector subspaces of (W, σ) where σ is an arbitrary symplectic form on W.

     

Sums of two squares in analytic rings

We study analytic singularities for which every positive semidefinite analytic function is a sum of two squares of analytic functions. This is a basic useful property of the plane, but difficult to check in other cases; in particular, what about , , or ? In fact, the unique positive examples we can find are the Brieskorn singularity, the union of two planes in 3-space and the Whitney umbrella. Conversely, we prove that a complete intersection with that property (other than the seven embedded surfaces already mentioned) must be a very simple deformation of the two latter, namely, In particular, except for the stems and , all singularities are real rational double points. Received April 4, 1997; in final form September 25, 1997     

On toric h-vectors of centrally symmetric polytopes

We prove tight lower bounds for the coefficients of the toric h-vector of an arbitrary centrally symmetric polytope generalizing previous results due to R. Stanley and the author using toric varieties. Our proof here is based on the theory of combinatorial intersection cohomology for normal fans of polytopes developed by G. Barthel, J.-P. Brasselet, K. Fieseler and L. Kaup, and independently by P. Bressler and V. Lunts. This theory is also valid for nonrational polytopes when there is no standard correspondence with toric varieties. In this way we can establish our bounds for centrally symmetric polytopes even without requiring them to be rational. Received: 24 March 2004     

Algebraic Equivalence and Homology Classes of Real Algebraic Cycles

In this note we study the spectral properties of a multiplication operator in the space L_p(X)^m which is given by an m by m matrix of measurable functions. Our particular interest is directed to the eigenvalues and the isolated spectral points which turn out to be eigenvalues. We apply these results in order to investigate the spectrum of an ordinary differential operator with so called “floating singularities”.     

The Behaviors of Expansion Functor on Monomial Ideals and Toric Rings

In this article, we study some algebraic and combinatorial behaviors of expansion functor. We show that on monomial ideals some properties like polymatroidalness, weakly polymatroidalness, and having linear quotients are preserved under taking the expansion functor.

The main part of the article is devoted to study of toric ideals associated to the expansion of subsets of monomials which are minimal with respect to divisibility. It is shown that, for a given discrete polymatroid P, if toric ideal of P is generated by double swaps, then toric ideal of any expansion of P has such a property. This result, in a special case, says that White's conjecture is preserved under taking the expansion functor. Finally, the construction of Gröbner bases and some homological properties of toric ideals associated to expansions of subsets of monomials is investigated.     

The Arithmetical Hierarchy of Real Numbers

A real number x is computable iff it is the limit of an effectively converging computable sequence of rational numbers, and x is left (right) computable iff it is the supremum (infimum) of a computable sequence of rational numbers. By applying the operations “sup” and “inf” alternately n times to computable (multiple) sequences of rational numbers we introduce a non‐collapsing hierarchy {Σ_n, Π_n, Δ_n : n ∈ ℕ} of real numbers. We characterize the classes Σ₂, Π₂ and Δ₂ in various ways and give several interesting examples.     

Subdivisions of Toric Complexes

We introduce toric complexes as polyhedral complexes consisting of rational cones together with a set of integral generators for each cone, and we define their associated face rings. Abstract simplicial complexes and rational fans can be considered as toric complexes, and the face ring for toric complexes extends Stanley and Reisner’s face ring for abstract simplicial complexes [20] and Stanley’s face ring for rational fans [21]. Given a toric complex with defining ideal I for the face ring we give a geometrical interpretation of the initial ideals of I with respect to weight orders in terms of subdivisions of the toric complex generalizing a theorem of Sturmfels in [23]. We apply our results to study edgewise subdivisions of abstract simplicial complexes.     

Characteristic exponents of semialgebraic singularities

Let X be a semialgebraic (or algebraic) set and let x₀ ∈ X be a singular point. There are some topological cycles of different dimensions contained in a small neighbourhood of x₀ in X. All these cycles vanish in x₀. The paper is devoted to “vanishing rates” of these cycles, which we call “characteristic exponents”. We prove that the characteristic exponents are invariant under bi‐Lipschitz transformations. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Rational Functions and Association Scheme Parameters

The parameters of metric, cometric, symmetric association schemes with q ± 1 (the same as the parameters of the underlying orthogonal polynomials) can be given in general by evaluating a single rational function of degree (4, 4) in the complex variable q ^j. But in all known examples, save the simple n-gons, these reduce to polynomials of degree at most 2 in q ^j with q an integer. One reason this occurs is that the rational function can have singularities at points which would determine some of the parameters. This paper deals with the case in which not all of the singularities are removable, thus giving some reason why the n-gons might naturally be the only exceptions to schemes with parameters being polynomials of degree at most 2 in q ^j , except possibly for schemes of very small diameter.     

The Hilbert scheme of points for supersingular abelian surfaces

We study the geometry of Hilbert schemes of points on abelian surfaces and Beauville’s generalized Kummer varieties in positive characteristics. The main result is that, in characteristic two, the addition map from the Hilbert scheme of two points to the abelian surface is a quasifibration such that all fibers are nonsmooth. In particular, the corresponding generalized Kummer surface is nonsmooth, and minimally elliptic singularities occur in the supersingular case. We unravel the structure of the singularities in dependence of p-rank and a-number of the abelian surface. To do so, we establish a McKay Correspondence for Artin’s wild involutions on surfaces. Along the line, we find examples of canonical singularities that are not rational singularities.