The Betti numbers of Mbar_{0,n}(r,d)

Contact Moishezon threefolds with second Betti number one

We prove that the only contact Moishezon threefold having second Betti number equal to one is the projective space.

Betti Numbers of Curves and Multiple-Point Loci

We construct Eagon--Northcott cycles on Hurwitz space and compare their classes to Kleiman's multiple point loci. Applying this construction towards the classification of Betti tables of canonical curves, we find that the value of the extremal Betti number records the number of minimal pencils. The result holds under transversality hypotheses equivalent to the virtual cycles having a geometric interpretation. We analyse the case of two minimal pencils, showing that the transversality hypotheses hold generically.

Milnor纤维的顶部Betti数的一个下界

A lower bound on the top Betti number of the Milnor fiber

本文给出了解析函数Milnor纤维顶部非零Betti数的下界，该下界由零维Lê数和临界轨迹复链接上消失循环的内部单值化决定。

Upper bounds on Betti numbers of tropical prevarieties

We prove upper bounds on the sum of Betti numbers of tropical prevarieties in dense and sparse settings. In the dense setting the bound is in terms of the volume of Minkowski sum of Newton polytopes of defining tropical polynomials, or, alternatively, via the maximal degree of these polynomials. In sparse setting, the bound involves the number of the monomials.

Virtual Betti numbers of real algebraic varieties

The weak factorization theorem for birational maps is used to prove that for all nonnegative i the ith mod 2 Betti number of compact nonsingular real algebraic varieties has a unique extension to a "virtual Betti number" beta_i defined for all real algebraic varieties, such that if Y is a closed subvariety of X, then beta_i(X) = beta_i(X\Y) + beta_i(Y).

Chow Rings of Mp_{0,2}(N,d) and Mbar_{0,2}(P^{N-1},d) and Gromov-Witten Invariants of Projective Hypersurfaces of Degree 1 and 2

In this paper, we prove formulas that represent two-pointed Gromov-Witten invariant <O_{h^a}O_{h^b}>_{0,d} of projective hypersurfaces with d=1,2 in terms of Chow ring of Mbar_{0,2}(P^{N-1},d), the moduli spaces of stable maps from genus 0 stable curves to projective space P^{N-1}. Our formulas are based on representation of the intersection number w(O_{h^a}O_{h^b})_{0,d}, which was introduced by Jinzenji, in terms of Chow ring of Mp_{0,2}(N,d), the moduli space of quasi maps from P^1 to P^{N-1} with two marked points. In order to prove our formulas, we use the results on Chow ring of Mbar_{0,…

Desingularized fiber products of semi-stable elliptic surfaces with vanishing third Betti number

Desingularized fiber products of semi-stable, non-isotrivial jacobian elliptic surfaces with vanishing third Betti number are classified. Such varieties may play a role in the study of supersingular threefolds, of the deformation theory of varieties with trivial canonical bundle, and of the arithmetic degenerations of rigid Calabi-Yau threefolds.

Deformation equivalence classes of complex surfaces with the first Betti number one, and the second Betti number zero

We will prove that the number of deformation equivalence classes of surfaces homotopy equivalent to a smooth, closed 4-manifold is finite, if the first Betti number is equal to one, and the second Betti number is equal to zero.

Lower bounds on Betti numbers

We survey recent results on bounds for Betti numbers of modules over polynomial rings, with an emphasis on lower bounds. Along the way, we give a gentle introduction to free resolutions and Betti numbers, and discuss some of the reasons why one would study these.

代数Betti数有多大？

About how large are algebraic Betti numbers?

本文利用Boij-Söderberg理论，为代数Betti数的大小提供了数量级上的界限估计。

The first $L^2$-Betti number for classifying spaces of variations of Hodge structures

The first Betti number for a lattice in a classifying space for variations of Hodge structures vanishes.

The first two Betti numbers of the moduli spaces of vector bundles on surfaces

We determined the first two Betti numbers of the moduli of rank two stable sheaves on an arbitrary algebraic surface

On the Betti numbers of compact holomorphic symplectic orbifolds of dimension four

We extend a result of Guan by showing that the second Betti number of a 4-dimensional primitively symplectic orbifold is at most 23 and there are at most 91 singular points. The maximal possibility 23 can only occur in the smooth case. In addition to the known smooth examples with second Betti numbers 7 and 23, we provide examples of such orbifolds with second Betti numbers 3, 5, 6, 8, 9, 10, 11, 14 and 16. In an appendix, we extend Salamon's relation among Betti/Hodge numbers of symplectic manifolds to symplectic orbifolds.

Upper Bounds for the Betti Numbers of a given Hilbert Function

From a Macaulay's paper it follows that a lex-segment ideal has the greatest number of generators (the 0-th Betti number $\b_0$) among all the homogeneous ideals with the same Hilbert function. In this paper we prove that this fact extends to every Betti number, in the sense that all the Betti numbers of a minimal free resolution of a lex segment ideal are bigger than or equal to the ones of any homogeneous ideal with the same Hilbert function.

Minimal free resolutions for homogeneous ideals with Betti numbers $1,n,n,1$

We investigate the standard generalized Gorenstein algebras of homological dimension three, giving a structure theorem for their resolutions. Moreover in many cases we are able to give a complete description of their graded Betti numbers.

Betti number bounds for fewnomial hypersurfaces via stratified Morse theory

We use stratified Morse theory for a manifold with corners to give a new bound for the sum of the Betti numbers of a hypersurface in R^n_> defined by a polynomial with n+l+1 terms.

On the Betti numbers of the tangent cones for Gorenstein Monomial Curves

The aim of the article is to study the Betti numbers of the tangent cone of Gorenstein monomial curves in affine 4-space. If $C_S$ is a non-complete intersection Gorenstein monomial curve whose tangent cone is Cohen-Macaulay, we show that the possible Betti sequences are (1,5,5,1), (1,5,6,2) and (1,6,8,3).

分次Betti数不是热带不变量

The graded Betti numbers are not tropical invariants

该研究证明分次Betti数不是齐次理想或赋值域上射影代数概形热带化的不变量，通过构造一组对应同一热带理想但Betti表不同的饱和理想完成论证。

The Betti numbers of regular Hessenberg varieties are palindromic

Recently Brosnan and Chow have proven a conjecture of Shareshian and Wachs describing a representation of the symmetric group on the cohomology of regular semisimple Hessenberg varieties for $GL_n(\mathbb{C})$. A key component of their argument is that the Betti numbers of regular Hessenberg varieties for $GL_n(\mathbb{C})$ are palindromic. In this paper, we extend this result to all reductive algebraic groups, proving that the Betti numbers of regular Hessenberg varieties are palindromic.

The algebraic dimension of compact complex threefolds with vanishing second Betti numbers

Small changes in sections 4 and 5, results not affected

Betti numbers for fat point ideals in the plane: a geometric approach

We consider the open problem of determining the graded Betti numbers for fat point subschemes supported at general points of the projective plane. We relate this problem to the open geometric problem of determining the splitting type of the pullback of the cotangent bundle on the plane to the normalization of certain rational plane curves. We give a conjecture for the graded Betti numbers which would determine them in all degrees but one for every fat point subscheme supported at general points of the plane. We also prove our Betti number conjecture in a broad range of cases. An appendix discu…

Betti numbers for certain Cohen-Macaulay tangent cones

In this article, we compute Betti numbers for a Cohen-Macaulay tangent cone of a monomial curve in the affine $4$-space corresponding to a pseudo symmetric numerical semigroup. As a byproduct, we also show that for these semigroups, being of homogeneous type and homogeneous are equivalent properties.

On the Total Curvature and Betti Numbers of Complex Projective Manifolds

We prove an inequality between the sum of the Betti numbers of a complex projective manifold and its total curvature, and we characterize the complex projective manifolds whose total curvature is minimal. These results extend the classical theorems of Chern and Lashof to complex projective space.

L^2-Betti numbers of plane algebraic curves

In [DJL07] it was shown that if A is an affine hyperplane arrangement in C^n, then at most one of the L^2-Betti numbers of its complement is non--zero. We will prove an analogous statement for complements of any algebraic curve in C^2. Furthermore we also recast and extend results of [LM06] in terms of L^2-Betti numbers.

The Betti Numbers of a Determinantal Variety

We determine the Poincaré polynomial of the determinantal variety $\{\det = 0\}$ in the projective space associated with the monoid of $n\times n$ matrices.

On the Betti numbers of Springer fibers for classical types

For a Weyl group $W$ of classical type, we present a formula to calculate the restriction of (graded) Springer representations of $W$ to a maximal parabolic subgroup $W'$ where the types of $W$ and $W'$ are in the same series. As a result, we obtain recursive formulas for the Betti numbers of Springer fibers for classical types.

Birational Calabi--Yau n-folds have equal Betti numbers

Let X and Y be two smooth projective n-dimensional algebraic varieties X and Y over C with trivial canonical line bundles. We use methods of p-adic analysis on algebraic varieties over local number fields to prove that if X and Y are birational, they have the same Betti numbers.

Intersection homology Betti numbers

A generalization of the formula of Fine and Rao for the ranks of the intersection homology groups of a complex algebraic variety is given. The proof uses geometric properties of intersection homology and mixed Hodge theory.

The second Betti number of hyperkähler manifolds

Let $M$ be a compact irreducible hyperkahler manifold, from Bogomolov inequality [V1] we obtain forbidden values of the second Betti number $b_2$ in arbitrary dimension. UPD: Unfortunately, decomposition of dual to BBF-form is not right in the main theorem. Instead of this work, take a look on recent preprints of Sawon and me on boundedness of $b_2$ for hyperkähler manifolds.

A Structure Theorem and the Graded Betti Numbers for Almost Complete Intersections

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.

Characteristic-independence of Betti numbers of graph ideals

In this paper we study the Betti numbers of Stanley-Reisner ideals generated in degree 2. We show that the first six Betti numbers do not depend on the characteristic of the ground field. We also show that, if the number of variables $n$ is at most 10, all Betti numbers are independent of the ground field. For $n=11$, there exists precisely 4 examples in which the Betti numbers depend on the ground field. This is equivalent to the statement that the homology of flag complexes with at most 10 vertices is torsion free and that there exists precisely 4 non-isomorphic flag complexes with 11 vertic…

非常面映射的多格Betti数

Multigraded Betti numbers of Veronese embeddings

本文研究非常面嵌入的多格Betti数，利用Hochster公式和离散Morse理论推导其消失与非消失结果。

Computing the First Betti Numberand Describing the Connected Components of Semi-algebraic Sets

In this paper we describe a singly exponential algorithm for computing the first Betti number of a given semi-algebraic set. Singly exponential algorithms for computing the zero-th Betti number, and the Euler-Poincaré characteristic, were known before. No singly exponential algorithm was known for computing any of the individual Betti numbers other than the zero-th one. We also give algorithms for obtaining semi-algebraic descriptions of the semi-algebraically connected components of any given real algebraic or semi-algebraic set in single-exponential time improving on previous results.

Betti numbers for numerical semigroup rings

We survey results related to the magnitude of the Betti numbers of numerical semigroup rings and of their tangent cones.

The Betti numbers of the moduli space of stable sheaves of rank 3 on P2

This article computes the generating functions of the Betti numbers of the moduli space of stable sheaves of rank 3 on the projective plane P2 and its blow-up. Wall-crossing is used to obtain the Betti numbers on the blow-up. These can be derived equivalently using flow trees, which appear in the physics of BPS-states. The Betti numbers for P2 follow from those for the blow-up by the blow-up formula. The generating functions are expressed in terms of modular functions and indefinite theta functions.