A fundamental tool in topological data analysis is persistent homology, which allows extraction of information from complex datasets a robust way. Persistent homology assigns module over principal ideal domain to one-parameter family spaces obtained the data. In applications, often depend on several parameters, and this case one interested studying multiparameter associated While theory for families well understood, situation more delicate. Following Carlsson Zomorodian, we recast problem...

We describe the minimal free resolution of ideal <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="2 times 2"> <mml:semantics> <mml:mrow> <mml:mn>2</mml:mn> <mml:mo>×</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">2 \times 2</mml:annotation> </mml:semantics> </mml:math> </inline-formula> subpermanents a n"> <mml:mi>n</mml:mi> n</mml:annotation> generic matrix alttext="upper M"> <mml:mi>M</mml:mi>...

The interplay between algebra and geometry is a beautiful (and fun!) area of mathematical investigation. Advances in computing algorithms make it possible to tackle many classical problems down-to-earth concrete fashion. This opens wonderful new vistas allows us pose, study solve that were previously out reach. Suitable for graduate students, the objective this 2003 book bring advanced life with lots examples. first chapters provide an introduction commutative connections geometry. rest...

We consider the problem of determining dimension space bivariate splines Ckr(Δ), for all k. This is closely related to question whether Cr(\̂gD) a free R-module. The main result that if and only ¦Δ¦ has genus zero Ckr(Δ) expected k = r + 1 (and hence k). also obtain several interesting corollaries, including following simple non-freeness criterion: given fixed Δ having an edge with both vertices interior, which does not extend boundary, there exists r0, can be determined by inspection, such any ≥ r0.

Let \mathcal C = \bigcup^n_{i 1} C_i ⊆ ℙ^2 be a collection of smooth rational plane curves. We prove that the addition–deletion operation used in study hyperplane arrangements has an extension which works for large class curves, giving inductive tool understanding freeness module Ω^1(\mathcal C) logarithmic differential forms with pole along . also show analog Terao’s conjecture (freeness is combinatorially determined if union lines) false this setting.

If $M$ is the complement of a hyperplane arrangement, and $A=H^*(M,\Bbbk )$ cohomology ring over field $\Bbbk$ characteristic $0$, then ranks, $\phi _k$, lower central series quotients $\pi _1(M)$ can be computed from Betti numbers, $b_{ii}=\dim \operatorname {Tor}^A_i(\Bbbk ,\Bbbk )_i$, linear strand in minimal free resolution $A$. We use Cartan-Eilenberg change rings spectral sequence to relate these numbers graded $b'_{ij}=\dim {Tor}^E_i(A,\Bbbk )_{j}$, $A$ exterior algebra $E$. From this...

Toric codes are evaluation obtained from an integral convex polytope $P \subset {\mathbb R}^n$ and finite field ${\mathbb F}_q$. They are, in a sense, natural extension of Reed–Solomon codes, have been studied recently [V. Diaz, C. Guevara, M. Vath, Proceedings Simu Summer Institute, 2001], [J. Hansen, Appl. Algebra Engrg. Comm. Comput., 13 (2002), pp. 289–300; Coding Theory, Cryptography Related Areas (Guanajuato, 1998), Springer, Berlin, 132–142], [D. Joyner, 15 (2004), 63–79]. In this...

Using multigraded Castelnuovo–Mumford regularity, we study the equations defining a projective embedding of variety syzygies are linear. This yields new results on toric varieties and normality polytopes.

The Orlik-Solomon algebra is the cohomology ring of complement a hyperplane arrangement A ⊆ C n ; it quotient an exterior Λ(V ) on |A| generators.In [9], Orlik and Terao introduced commutative analog Sym(V * )/I to answer question Aomoto showed Hilbert series depends only intersection lattice L(A).In [6], Falk Randell define property 2-formality; in this note we study relation between 2-formality Orlik-Terao algebra.Our main result necessary sufficient condition for 2formality terms...

In a recently published paper [Trans. Amer. Math. Soc. 363 (2011) 229–257], Migliore, Miró-Roig and Nagel show that the weak Lefschetz property (WLP) can fail for an ideal I ⊆ 𝕜[x1, …, x4] generated by powers of linear forms. This is in contrast to analogous situation x2, x3], where WLP always holds [H. Schenck A. Seceleanu, Proc. 138 (2010) 2335–2339]. We use inverse system dictionary connect fat points, failure forms connected geometry associated point scheme. Recent results Sturmfels Xu...

We define a complex R/J of graded modules on ad-dimensional simplicial Δ, so that the top homology module this consists piecewise polynomial functions (splines) smoothnessron cone Δ. In series papers,4;5;6] used similar approach to study dimension spaces splines but with substantially different from R/J. obtain bounds modulesHi(R/J) for alli < dand find spectral sequence which relates these spline module. use give simple conditions governing projective also prove if is free, then it...

Journal Article Resonance Varieties Via Blowups of ℙ2 and Scrolls Get access Hal Schenck Department Mathematics, University Illinois, Urbana, IL 61801, USA Correspondence to be sent to: e-mail: schenck@math.uiuc.edu Search for other works by this author on: Oxford Academic Google Scholar International Mathematics Research Notices, Volume 2011, Issue 20, Pages 4756–4778, https://doi.org/10.1093/imrn/rnq271 Published: 17 December 2010 history Received: 27 June Revision received: 14 November...

For a toric variety $X_\Sigma$ determined by polyhedral fan $\Sigma \subseteq N$, Payne shows that the equivariant Chow cohomology is $\mathrm {Sym}(N)$-algebra $C^0(\Sigma )$ of integral piecewise polynomial functions on $\Sigma$. We use Cartan-Eilenberg spectral sequence to analyze associated reflexive sheaf $\mathcal {C}^0(\Sigma $\mathbb {P}_{\mathbb {Q}}(N)$, showing Chern classes depend subtle geometry $\Sigma$ and giving criteria for splitting as sum line bundles. certain fans...

For a simplicial subdivision Δ of region inR2, we analyze the dimension vector spaceCkr(Δ) ofCrpiecewise polynomial functions (splines) on degree at mostk. We find an exact sequence which allows us to prove that series for splines given by5does indeed agree with bounds spline space by Alfeld and Schumaker [1;2]. give sufficient conditions Alfeld–Schumaker be attained in all degrees, where is two-dimensional complex. The are satisfied class complexes considered by6, but also much broader...

If $\mathcal A$ is a complex hyperplane arrangement, with complement $X$, we show that the Chen ranks of $G=\pi _1(X)$ are equal to graded Betti numbers linear strand in minimal, free resolution cohomology ring $A=H^*(X,\Bbbk )$, viewed as module over exterior algebra $E$ on A$: \[ \theta _k(G) = \dim _{\Bbbk }\operatorname {Tor}^E_{k-1}(A,\Bbbk )_k, \quad \text {for $k\ge 2$}, \] where $\Bbbk$ field characteristic $0$. The conjecture asserts that, for $k$ sufficiently large, $\theta =(k-1)...

We show that an Artinian quotient of ideal <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper I subset-of-or-equal-to double-struck upper K left-bracket x comma y z right-bracket"> <mml:semantics> <mml:mrow> <mml:mi>I</mml:mi> <mml:mo>⊆</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">K</mml:mi> </mml:mrow> <mml:mo stretchy="false">[</mml:mo> <mml:mi>x</mml:mi> <mml:mo>,</mml:mo>...

Associated to an $n$-dimensional integral convex polytope $P$ is a toric variety $X$ and divisor $D$, such that the points of represent $H^0({\mathcal O}_X(D))$. We study free resolution homogeneous coordinate ring $\bigoplus _{m \in \mathbb Z}H^0(mD)$ as module over $Sym(H^0({\mathcal O}_X(D)))$. It turns out simple application Green’s theorem yields good bounds for linear syzygies projective surface. In particular, planar $P=H^0({\mathcal O}_X(D))$, $D$ satisfies condition $N_p$ if...