Searcharxiv⌕ Search
Home/Archive/arXiv

arXiv · 0706.2588

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

Alessandro Gimigliano·Brian Harbourne·Monica Idà
View original source ↗Read source document ↗Export metadata ↓

Abstract

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 discusses many more cases in which our conjecture has been verified computationally and provides a new and more efficient computational approach for computing graded Betti numbers in certain degrees. It also demonstrates how to derive explicit conjectural values for the Betti numbers and how to compute splitting types.

Explore related subjects

math.AGmath.AC

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Alessandro Gimigliano, Brian Harbourne, Monica Idà. 2007-06-18. Betti numbers for fat point ideals in the plane: a geometric approach. https://arxiv.org/abs/0706.2588

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Categorifying Quiver Linking/Unlinking using CoHA Modules

The knots-quivers correspondence is a relation between knot invariants and enumerative invariants of quivers, which in particular translates the knot operations of linking and unlinking to a certain mutation operation on quivers. In this paper we show that the moduli spaces of a quiver and its linking/unlinking are naturally related, giving a purely representation theoretic interpretation of these operations. We obtain a relation between the cohomologies of these spaces which is moreover compatible with a natural action of the Cohomological Hall Algebra. The result is a categorification of quiver linking/unlinking at the level of CoHA modules.

math.AG↗
arXiv2026

On the prime ideals of higher secant varieties of Veronese embeddings of small degrees

In this paper, we study minimal generators of the (saturated) defining ideal of the $k$-secant variety $σ_k(v_d(\mathbb{P}^n))$ of the image of the $d$-uple Veronese embedding $v_d: \mathbb{P}^n \rightarrow \mathbb{P}^N$ with ${N=\binom{n+d}{d}-1}$, focusing on cases where the degree of $σ_k(v_d(\mathbb{P}^n))$ is relatively small. First, we show that the prime ideal $I(σ_4(v_3(\mathbb{P}^3)))$ is minimally generated by $36$ homogeneous polynomials of degree $5$. This implies that $σ_4(v_3(\mathbb{P}^3)) \subset \mathbb{P}^{19}$ is a del Pezzo $4$-secant variety (i.e., $\mathrm{deg}(σ_4(v_3(\mathbb{P}^3))) = 105$ and the sectional genus $π(σ_4(v_3(\mathbb{P}^3))) = 316$), thereby providing a new example of an arithmetically Gorenstein variety of codimension $4$. This result addresses the symmetric version of the ``Salmon problem'' posed by E. Allman in \cite{Allman}. As an application, we decide the non-singularity of a certain locus in $σ_4(v_3(\mathbb{P}^3))$. Furthermore, by inheritance, we obtain the generators of $I(σ_4(v_3(\mathbb{P}^n)))$ for all $n \geq 3$. Based on the method used for $σ_4(v_3(\mathbb{P}^3))$, we also propose a procedure to compute the first non-trivial degree piece, $I(σ_k(v_d(\mathbb{P}^n)))_{k+1}$, for the general $k$-secant case using prolongation and weight space decomposition. Applying this procedure, we present a few more cases of $k$-secant varieties of relatively small degrees; in each of these cases, the ideal is generated in degree $k+1$ and can be fully determined by explicitly computing all generators within this degree piece.

math.AG↗
arXiv2026

Symmetric quasi-coherent sheaves

Using methods of stable homotopy theory, the category of symmetric quasi-coherent sheaves associated with non-commutative graded algebras with extra symmetries is introduced and studied in this paper. It is shown to be a closed symmetric monoidal Grothendieck category with invertible generators. It is proven that the category of quasi-coherent sheaves on a projective scheme is recovered out of symmetric quasi-coherent sheaves. As an application, symmetric projective schemes associated to such algebras are introduced and studied. It is shown that classical projective schemes are recovered from symmetric ones.

math.AG↗