Monomial invariants in codimension two

Standard monomial bases and geometric consequences for certain rings of invariants

Consider the diagonal action of $SL_n(K)$ on the affine space $X=V^{\oplus m}\oplus (V^*)^{\oplus q}$ where $V=K^n, K$ an algebraically closed field of arbitrary characteristic and $m,q>n$. We construct a "standard monomial" basis for the ring of invariants $K[X]^{SL_n(K)}$. As a consequence, we deduce that $K[X]^{SL_n(K)}$ is Cohen-Macaulay. We also present the first and second fundamental theorems for $SL_n(K)$-actions.

The vanishing of a higher codimension analog of Hochster's theta invariant

We study H. Dao's invariant $η_c^R$ of pairs of modules defined over a complete intersection ring $R$ of codimension $c$ having an isolated singularity. Our main result is that $η_c^R$ vanishes for all pairs of modules when $R$ is a {\em graded} complete intersection ring of codimension $c > 1$ having an isolated singularity. A consequence of this result is that all pairs of modules over such a ring are $c$-$\Tor$-rigid.

Roots of Bernstein-Sato polynomials for monomial ideals: a positive characteristic approach

We describe the roots of the Bernstein-Sato polynomial of a monomial ideal using reduction mod p and invariants of singularities in positive chracteristic. We give in this setting a positive answer to a problem of Takagi, Watanabe and the second author, concerning the dependence on the characteristic for these invariants of singularities.

Standard monomial bases, moduli of vector bundles, and invariant theory

Consider the diagonal action of the special orthogonal group on the direct sum of a finite number of copies of the standard representation--the underlying field is assumed to be algebraically closed and of characteristic not equal to two. We construct a "standard monomial" basis for the ring of polynomial invariants for this action. We then deduce, by a deformation argument, our main result that this ring of polynomial invariants is Cohen-Macaulay. We give three applications of this result: (1) the first and second fundamental theorems of invariant theory for the above action; (2) Cohen-Macaul…

弧空间的标准单项式与不变量理论 II：辛群

Standard monomials and invariant theory of arc spaces II: Symplectic group

本文为任意代数闭域上 Pfaffian 簇的弧空间构造标准单项式基，并由此证明辛群不变量理论第一、第二基本定理的弧空间版本。

Higher Codimensional Alpha Invariants and Characterization of Projective Spaces

We generalize the definition of alpha invariant to arbitrary codimension. We also give a lower bound of these alpha invariants for K-semistable Q-Fano varieties and show that we can characterize projective spaces among all K-semistable Fano manifolds in terms of higher codimensional alpha invariants. Our results demonstrate the relation between alpha invariants of any codimension and volumes of Fano manifolds in the characterization of projective spaces.

关于α-和δ-不变量的准单变量性

On the quasi-monomiality of the $α$- and $δ$-invariants

本文研究了在代数闭域上，对于任何准正的有理极子L，α和δ不变量通过准单变量计算，无需假设基础域的不可数性。

Algebraic invariants of projective monomial curves associated to generalized arithmetic sequences

Let $K$ be an infinite field and let $m_1,\ldots,m_n$ be a generalized arithmetic sequence of positive integers, i.e., there exist $h, d, m_1 \in\mathbb{Z}^+$ such that $m_i = h m_1 + (i-1)d$ for all $i \in \{2,\ldots,n\}$. We consider the projective monomial curve $\mathcal C\subset \mathbb{P}^{n}_{K}$ parametrically defined by $$x_1=s^{m_1}t^{m_n-m_1},\dots,x_{n-1}=s^{m_{n-1}}t^{m_n-m_{n-1}},x_n=s^{m_n},x_{n+1}=t^{m_n}.$$ In this work, we characterize the Cohen-Macaulay and Koszul properties of the homogeneous coordinate ring $K[\mathcal C]$ of $\mathcal C$. Whenever $K[\mathcal C]$ is Cohen…

Separating monomials for diagonalizable actions

Sets of monomials separating Zariski closed orbits under diagonalizable group actions are characterized in terms of the monoid of zero-sum sequences over the character group. This is applied to compare the degree bounds for separating invariants and generating invariants of diagonalizable group actions.

弧空间的标准单项式与不变量理论 III：特殊线性群

Standard monomials and invariant theory for arc spaces III: special linear group

本文证明特殊线性群不变量理论第一与第二基本定理的弧空间类比，指出其因仿射商弧空间可能非既约而比一般线性群与辛群情形更微妙。

Dynamical invariants of monomial correspondences

We focus on various dynamical invariants associated to toric correspondences, using algebraic geometry or arithmetic. We find a formula for the dynamical degrees, relate the exponential growth of the degree sequences with a strict log-concavity condition on the dynamical degrees and compute the asymptotic ratio of the growth of heights of points of such correspondences.

Hypersurfaces quasi-invariant by codimension one foliations

We present a variant of the classical Darboux-Jouanolou Theorem. Our main result provides a characterization of foliations which are pull-backs of foliations on surfaces by rational maps. As an application, we provide a structure theorem for foliations on 3-folds admitting an infinite number of extremal rays.

A new strategy for resolution of singularities in the monomial case in positive characteristic

According to our approach for resolution of singularities in positive characteristic (called the Idealistic Filtration Program, alias the I.F.P. for short) the algorithm is devided into the following two steps: Step 1. Reduction of the general case to the monomial case. Step 2. Solution in the monomial case. While we have established Step 1 in arbitrary dimension, Step 2 becomes very subtle and difficult in positive characteristic. This is in clear contrast to the classical setting in characteristic zero, where the solution in the monomial case is quite easy. In dimension 3, we provided an inv…

Liaison invariants and the Hilbert scheme of codimension 2 subschemes in P^{n+2}

In this paper we study the Hilbert scheme, Hilb(P), of equidimensional locally Cohen-Macaulay codimension 2 subschemes, with a special look to surfaces in P^4 and 3-folds in P^5, and the Hilbert scheme stratification H_{c} of constant cohomology. For every (X) in Hilb(P) we define a number δ_X in terms of the graded Betti numbers of the homogeneous ideal of X and we prove that 1 + δ_X - \dim_{(X)} H_{c} and 1 + δ_X - \dim T_{c} are CI-biliaison invariants where T_{c} is the tangent space of H_{c} at (X). As a corollary we get a formula for the dimension of any generically smooth component of H…

Integral Subschemes of Codimension Two

In this paper we study the problem of describing the integral subschemes within a fixed even linkage class $Ł$ of subschemes in $\Pn$ of codimension two. In the case that $Ł$ is not the class of arithmetically Cohen-Macaulay subschemes, we associate to any $X \in Ł$ two invariants $θ_X$ and $η_X$. When taken with the height $h_X$, each of these invariants determines the location of $X$ in $Ł$, thought of as a poset under domination. In terms of these invariants, necessary conditions are given for integral subschemes. The necessary conditions are almost sufficient in the sense that if a subsche…

The hyperdeterminant of 3 x 3 x 2 arrays, and the simplest invariant of 4 x 4 x 2 arrays

We use the representation theory of Lie algebras and computational linear algebra to obtain an explicit formula for the hyperdeterminant of a $3 \times 3 \times 2$ array: a homogeneous polynomial of degree 12 in 18 variables with 16749 monomials and 41 distinct integer coefficients; the monomials belong to 178 orbits under the action of $(S_3 \times S_3 \times S_2) \rtimes S_2$. We also obtain the simplest invariant for a $4 \times 4 \times 2$ array: a homogeneous polynomial of degree 8 in 32 variables with 14148 monomials and 13 distinct integer coefficients; the monomials belong to 28 orbits…

Level algebras with bad properties

This paper can be seen as a continuation of the works contained in the recent preprints [Za], of the second author, and [Mi], of Juan Migliore. Our results are: 1). There exist codimension three artinian level algebras of type two which do not enjoy the Weak Lefschetz Property (WLP). In fact, for $e\gg 0$, we will construct a codimension three, type two $h$-vector of socle degree $e$ such that {\em all} the level algebras with that $h$-vector do not have the WLP. We will also describe the family of those algebras and compute its dimension, for each $e\gg 0$. 2). There exist reduced level sets …

Codimension one distributions of degree 2 on the three-dimensional projective space

We establish a full classification of degree $2$ codimension one distributions on $\mathbb{P}^3$ according to invariants of their tangent sheaves.

Monoidal transforms and invariants of singularities in positive characteristic

The problem of resolution of singularities in positive characteristic can be reformulated as follows: Fix a hypersurface $X$, embedded in a smooth scheme, with points of multiplicity at most $n$. Let an $n$-sequence of transformations of $X$ be a finite composition of monoidal transformations with centers included in the $n$-fold points of $X$, and of its successive strict transforms. The open problem (in positive characteristic) is to prove that there is an $n$-sequence such that the final strict transform of $X$ has no points of multiplicity $n$ (no $n$-fold points). In characteristic zero, …

Polar degrees and closest points in codimension two

Suppose that $X_A\subset \mathbb{P}^{n-1}$ is a toric variety of codimension two defined by an $(n-2)\times n$ integer matrix $A$, and let $B$ be a Gale dual of $A$. In this paper we compute the Euclidean distance degree and polar degrees of $X_A$ (along with other associated invariants) combinatorially working from the matrix $B$. Our approach allows for the consideration of examples that would be impractical using algebraic or geometric methods. It also yields considerably simpler computational formulas for these invariants, allowing much larger examples to be computed much more quickly than…

Cup products, the Heisenberg group, and codimension two algebraic cycles

We define higher categorical invariants (gerbes) of codimension two algebraic cycles and provide a categorical interpretation of the intersection of divisors on a smooth proper algebraic variety. This generalization of the classical relation between divisors and line bundles furnishes a new perspective on the Bloch-Quillen formula.

The Generalized Hodge conjecture for 1-cycles and codimension two algebraic cycles

In this paper, we prove that the statement: ``The (Generalized) Hodge Conjecture holds for codimension-two cycles on a smooth projective variety $X$" is a birationally invariant statement, that is, if the statement is true for $X$, it is also true for all smooth varieties $X'$ which are birationally equivalent to $X$. We also prove the analogous result for 1-cycles. As direct corollaries, the Hodge Conjecture holds for smooth rational projective manifolds with dimension less than or equal to five, and, the Generalized Hodge Conjecture holds for smooth rational projective manifolds with dimensi…

Branched pull-back components of the space of codimension 1 foliations on $\mathbb P^n$

Let $\mathcal{F}$ be written as $ f^{*}\mathcal{G}$, where $\mathcal{G}$ is a foliation in $ {\mathbb P^2}$ with three invariant lines in general position, say $(XYZ)=0$, and $f:{\mathbb P^n}--->{\mathbb P^2}$, $f=(F^α_{0}:F^β_{1}:F^γ_{2})$ is a nonlinear rational map. Using local stability results of singular holomorphic foliations, we prove that: if $n\geq 3$, the foliation $\mathcal{F}$ is globally stable under holomorphic deformations. As a consequence we obtain new irreducible componentes for the space of codimension one foliations on $\mathbb P^n$. We present also a result which characte…