Margherita Barile
We present a class of toric varieties $V$ which, over any algebraically closed field of characteristic zero, are defined by codim $V$+1 binomial equations.
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CI-sequences and almost complete intersections
本文研究几乎完全交阿廷代数的希尔伯特函数与分次贝蒂数，证明完全交的希尔伯特函数均可由几乎完全交实现，并在余维数3情形下分类了可能的贝蒂数与极小自由消解中的消去。
We provide a structure theorem for all almost complete intersection ideals of depth three in any Noetherian local ring. In particular, we find that the minimal generators are the pfaffians of suitable submatrices of an alternating matrix. From the graded version of the previous result, we characterize the graded Betti numbers of all 3-codimensional almost complete intersection schemes of P^r.
We describe a class of affine toric varieties $V$ that are set-theoretically minimally defined by codim $V+1$ binomial equations over fields of any characteristic.
Many algebras are expected to have the Weak Lefschetz property though this is often very difficult to establish. We illustrate the subtlety of the problem by studying monomial and some closely related ideals. Our results exemplify the intriguing dependence of the property on the characteristic of the ground field, and on arithmetic properties of the exponent vectors of the monomials.
A current research theme is to compare symbolic powers of an ideal $I$ with the regular powers of $I$. In this paper, we focus on the case that $I=I_X$ is an ideal defining an almost complete intersection (ACI) sets of points $X$ in $\mathbb{P}^1\times\mathbb{P}^1$. In particular, we describe a minimal free bigraded resolution of a non arithmetically Cohen-Macaulay (also non homogeneus) set of fat points $\mathcal Z$ whose support is an ACI. We call $\mathcal Z$ a fat ACI. We also show that its symbolic and ordinary powers are equal, i.e, $I_{\mathcal Z}^{(m)}=I_{\mathcal Z}^{m}$ for any $m\ge…
We generalize some properties related to Hilbert series and Lefschetz properties of Milnor algebras of projective hypersurfaces with isolated singularities to the more general case of an almost complete intersection ideal $J$ of dimension one. When the saturation $I$ of $J$ is a complete intersection, we get explicit formulas for a number of related invariants. New examples of hypersurfaces $V:f=0$ in $P^n$ whose Jacobian ideal $J_f$ satisfies this property and with explicit nontrivial Alexander polynomials are given in any dimension. A Lefschetz type property for the graded quotient $I/J$ is …
It is known that all complete intersection Artinian standard graded algebras of codimension 3 have the Weak Lefschetz Property. Unfortunately, this property does not continue to be true when you increase the number of minimal generators for the ideal defining the algebra. For instance, it is not more valid for almost complete intersection Artinian standard graded algebras of codimension 3. On the other hand, the Hilbert functions of all Weak Lefschetz Artinian graded algebras are unimodal and with the positive part of their first difference forming an $O$-sequence (i.e. are Weak Lefschetz sequ…
We study properties of the resolution of almost Gorenstein artinian algebras $R/I,$ i.e. algebras defined by ideals $I$ such that $I=J+(f),$ with $J$ Gorenstein ideal and $f\in R.$ Such algebras generalize the well known almost complete intersection artinian algebras. Then we give a new explicit description of the resolution and of the graded Betti numbers of almost complete intersection ideals of codimension $3$ and we characterize the ideals whose graded Betti numbers can be achieved using artinian monomial ideals.
Using the connections among almost complete intersection schemes, arithmetically Gorenstein schemes and schemes that are union of complete intersections we give a structure theorem for arithmetically Cohen-Macaulay union of two complete intersections of codimension $2.$ We apply the results for computing Hilbert functions and graded Betti numbers for such schemes.
We give a generalization of the duality of a zero-dimensional complete intersection to the case of one-dimensional almost complete intersections, which results in a {\em Gorenstein module} $M=I/J$. In the real case the resulting pairing has a signature, which we show to be constant under flat deformations. In the special case of a non-isolated real hypersurface singularity $f$ with a one-dimensional critical locus, we relate the signature on the jacobian module $I/J_f$ to the Euler characteristic of the positive and negative Milnor fibre, generalising the result for isolated critical points. A…
It is proved in this paper that a locally complete intersection curve in a smooth affine C-algebra with trival conormal bundle is a set theoretic complete intersection if its corresponding class in the Grothendieck Group is torsion.
This paper has been withdrawn by the auther due to an error
In this paper we produce infinitely many examples of set-theoretic complete intersection monomial curves in $\mathbb{P}^{n+1}$, starting with a set-theoretic complete intersection monomial curve in $\mathbb{P}^{n}$ . In most of the cases our results cannot be obtained through semigroup gluing technique and we can tell apart explicitly which cases are new.
Using aritmethic conditions on affine semigroups we prove that for a simplicial toric variety of codimension 2 the property of being a set-theoretic complete intersection on binomials in characteristic $p$ holds either for all primes $p$, or for no prime $p$, or for exactly one prime $p$.
We present a class of homogeneous ideals which are generated by monomials and binomials of degree two 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 two.
Based on the fact that projective monomial curves in the plane are complete intersections, we give an effective inductive method for creating infinitely many monomial curves in the projective $n$-space that are set theoretic complete intersections. We illustrate our main result by giving different infinite families of examples. Our proof is constructive and provides one binomial and $(n-2)$ polynomial explicit equations for the hypersurfaces cutting out the curve in question.
The goal of this paper is to study a $p$-adic analog of the joint of the conjectures of André--Oort and André--Pink. More precisely, on a product of ordinary Siegel formal moduli schemes, we study the distribution of points whose components are either CM points or points in Hecke orbits. We use linearity of formal subschemes of the product as the $p$-adic analog of geodesicness over complex numbers. Moreover, we relax the usual incidence relations by using $p$-adic distance. We also study a $p$-adic formal scheme theoretic analog of the Ax--Lindemann theorem.
The arithmetic rank of the residual intersections of a complete intersection ideal
研究完全交理想残差交的算术秩，确定其在任意特征下的上界，并给出其集合理论生成元的显式描述。
Let $\mathcal A\subset\mathbb P^{k-1}$ be a rank $k$ arrangement of $n$ hyperplanes, with the property that any $k$ of the defining linear forms are linearly independent (i.e., $\mathcal A$ is called $k-$generic). We show that for any $j=0,\ldots,k-2$, the subspace arrangement with defining ideal generated by the $(n-j)-$fold products of the defining linear forms of $\mathcal A$ is a set-theoretic complete intersection, which is equivalent to saying that star configurations have this property.
In this paper we completely characterize lattice ideals that are complete intersections or equivalently complete intersections finitely generated semigroups of $\bz^n\oplus T$ with no invertible elements, where $T$ is a finite abelian group. We also characterize the lattice ideals that are set-theoretic complete intersections on binomials.
We deform monomial space curves in order to construct examples of set-theoretical complete intersection space curve singularities. As a by-product we describe an inverse to Herzog's construction of minimal generators of non-complete intersection numerical semigroups with three generators.
We prove a Cayley-Bacharach-type theorem for points in projective space $\mathbb{P}^n$ that lie on a complete intersection of $n$ hypersurfaces. This is made possible by new bounds on the growth of the Hilbert function of almost complete intersections.
Homotopy continuation provides a numerical tool for computing the equivalence of a smooth variety in an intersection product. Intersection theory provides a theoretical tool for relating the equivalence of a smooth variety in an intersection product to the degrees of the Chern classes of the variety. A combination of these tools leads to a numerical method for computing the degrees of Chern classes of smooth projective varieties in P^n. We illustrate the approach through several worked examples.
Using the Grothendieck-Lefschetz theory (see \cite{[SGA2]}) we prove a criterion to deduce that certain subvarieties of $\mathbb P^n$ of dimension $\geq 2$ are not set-theoretic complete intersections (see Theorem 1 of the Introduction). As applications we give a number of relevant examples. In the last part of the paper we prove that the arithmetic rank of a rational normal $d$-dimensional scroll $S_{n_1,...,n_d}$ in $\mathbb P^N$ is $N-2$, by producing an explicit set of $N-2$ homogeneous equations which define these scrolls set-theoretically (see Theorem 2 of the Introduction).
A Deformation Theoretic Reduction of the Hodge Conjecture via Derived Categories and Complete Intersections
本文提出几何与范畴方法，通过嵌入平坦族并利用导出范畴证明完全交情形的霍奇猜想，并将一般情形约化为极限代数性假设，统一了霍奇理论、形变与导出技术。
We characterize transversality, non-transversality properties on the moduli space of genus 0 stable maps to a semipositive symplectic manifold of dimension 4, when GW([point],...,[point]) is enumerative. In particular, we show that the intersection theoretic property depends on the existence of a critical point on a stable map.
Let K be an algebraically closed field of characteristic zero and let I=(f_1,...,f_n) be a homogeneous R_+-primary ideal in R:=K[X,Y,Z]. If the corresponding syzygy bundle Syz(f_1,...,f_n) on the projective plane is semistable, we show that the Artinian algebra R/I has the Weak Lefschetz property if and only if the syzygy bundle has a special generic splitting type. As a corollary we get the result of Harima et alt., that every Artinian complete intersection (n=3) has the Weak Lefschetz property. Furthermore, we show that an almost complete intersection (n=4) does not necessarily have the Weak…
Let $X$ be an isotropic Grassmannian of type $B$, $C$, or $D$. In this paper we calculate $K$-theoretic Pieri-type triple intersection numbers for $X$: that is, the sheaf Euler characteristic of the triple intersection of two arbitrary Schubert varieties and a special Schubert variety in general position. We do this by determining explicit equations for the projected Richardson variety corresponding to the two arbitrary Schubert varieties, and show that it is a complete intersection in projective space. The $K$-theoretic Pieri coefficients are alternating sums of these triple intersection numb…
Given a projective morphism of compact, complex, algebraic varieties and a relatively ample line bundle on the domain we prove that a suitable choice, dictated by the line bundle, of the decomposition isomorphism of the Decomposition Theorem of Beilinson, Bernstein, Deligne and Gabber, yields isomorphisms of pure Hodge structures. The proof is based on a new cohomological characterization of the decomposition isomorphism associated with the line bundle. We prove some corollaries concerning the intersection form in intersection cohomology, the natural map from cohomology to intersection cohomol…
A Hodge Theoretic Generalization of $\mathbb{Q}$-Homology Manifolds II: Local Complete Intersections
研究复代数簇$Z$的奇点不变量$\HRH(Z)$，聚焦局部完全交子簇，将其与伯恩斯坦 - 佐藤多项式等著名不变量联系，超曲面情形可由这些不变量完全刻画$\HRH(Z)$，高余维情形更复杂。
A multiple (loc. Cohen Macaulay) structure, X, on a space curve C in P3 is said to be primitive if X is locally contained in a smooth surface. We give numerical conditions for C to be a "primitive" set theoretic complete intersection (i.e. to have a primitive multiple structure which is a complete intersection).
The interpretation, due to T. Mabuchi, of the classical Futaki invariant of Fano toric manifolds is extended to the case of the Generalized Futaki invariant, introduced by W. Ding and G. Tian, of almost Fano toric varieties. As an application it is shown that the real part of the Generalized Futaki invariant is positive for all degenerations of the Fano manifold V_{38}, obtained by intersection of the Veronese embedding of ${\bf P}^3\times{\bf P}^2 \subset {\bf P}^{11}$ with codimension-two hyperplanes.
Our main theorem characterizes the complete intersections of codimension 2 in a projective space of dimension 3 or more over an algebraically closed field of characteristic 0 as the subcanonical and self-linked subschemes. In order to prove this theorem, we'll prove the Gherardelli linkage theorem, which asserts that a partial intersection of two hypersurfaces is subcanonical if and only if its residual intersection is, scheme-theoretically, the intersection of the two hypersurfaces with a third.
We study intersection theoretic problems in the setting of Chow-Witt groups with coefficients in a fixed Milnor-Witt cycle algebra over a perfect field. We prove that the product maps on such groups satisfy the following property: given two points in a regular local scheme with supports which do not intersect properly, their product vanishes. This gives an analogue of Serre's vanishing result for intersection multiplicities.
We prove a K-theoretic excess intersection formula for derived Artin stacks. When restricted to classical schemes, it gives a refinement and new proof of R. Thomason's formula.
We prove in full generality the mirror duality conjecture for string-theoretic Hodge numbers of Calabi-Yau complete intersections in Gorenstein toric Fano varieties. The proof is based on properties of intersection cohomology
We show that under favorable circumstances, one can construct an intersection product on the Chow groups of a tensor triangulated category $\mathcal{T}$ (as defined by Balmer) which generalizes the usual intersection product on a non-singular algebraic variety. Our construction depends on the choice of an algebraic model for $\mathcal{T}$ (a tensor Frobenius pair), which has to satisfy a $\mathrm{K}$-theoretic regularity condition analogous to the Gersten conjecture from algebraic geometry. In this situation, we are able to prove an analogue of the Bloch formula and use it to define an interse…
We provide an intersection-theoretic formula for the Euler characteristic of the moduli space of smooth curves. This formula reads purely in terms of Hodge integrals and, as a corollary, the standard calculus of tautological classes gives a new short proof of the Harer-Zagier formula. Our result is based on the Gauss-Bonnet formula, and on the observation that a certain parametrisation of the $Ω$-class - the Chern class of the universal $r$-th root of the twisted log canonical bundle - provides the Chern class of the log tangent bundle to the moduli space of smooth curves. Being $Ω$-classes by…
Intersection theoretic inequalities via Lorentzian polynomials
该研究将洛伦兹多项式应用于多类几何领域，建立了rKT性质等相交理论不等式与凸性几何不等式，拓展了多拟阵结构适用范围并得到加性组合不等式的几何类似结果。
Let S, T be surfaces in P3. Suppose that S intersect T is set-theoretically a smooth curve C of degree d and genus g. Suppose that S and T have no common singular points. Then if C is not a complete intersection, then deg(S), deg(T) < 2d^4. Fixing (d,g), one can form a finite (shorter) list of all possible pairs (deg(S),deg(T)). For instance, when (d,g) = (4,0), and assuming for simplicity that deg(S) <= deg(T): (deg(S), deg(T)) \in {(3,4), (3,8), (4,4), (4,7), (6,26), (9,48), (10,28) (12,18), (13,16), (17,220), (18,118), (19,84), (20,67), (22,50), (28,33)}. Assume characteristic 0. [1] Suppos…
This paper deals with properties of the algebraic variety defined as the set of zeros of a "typical" sequence of polynomials. We consider various types of "nice" varieties: set-theoretic and ideal-theoretic complete intersections, absolutely irreducible ones, and nonsingular ones. For these types, we present a nonzero "obstruction" polynomial of explicitly bounded degree in the coefficients of the sequence that vanishes if its variety is not of the type. Over finite fields, this yields bounds on the number of such sequences. We also show that most sequences (of at least two polynomials) define…
We investigate Hodge-theoretic properties of Calabi-Yau complete intersections $V$ of $r$ semi-ample divisors in $d$-dimensional toric Fano varieties having at most Gorenstein singularities. Our main purpose is to show that the combinatorial duality proposed by second author agrees with the duality for Hodge numbers predicted by mirror symmetry. It is expected that the complete verification of mirror symmetry predictions for singular Calabi-Yau varieties $V$ of arbitrary dimension demands considerations of so called {\em string-theoretic Hodge numbers} $h^{p,q}_{\rm st}(V)$. We restrict oursel…
Irrational families of Fano weighted complete intersection fourfolds
该研究基于格罗莫夫-维滕不变量的原子论方法，证明了加权射影空间中广泛族光滑法诺完全交四元重流形的辛非有理性，填补了此前仅对一般成员成立的结论空白。
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