A Note on the Multiplicity of Cohen-Macaulay Algebras with Pure Resolutions

1985 ◽  
Vol 37 (6) ◽  
pp. 1149-1162 ◽  
Author(s):  
Craig Huneke ◽  
Matthew Miller
Keyword(s):  
Tangent Cone ◽  
Betti Numbers ◽  
Rational Surface ◽  
Minimal Resolution ◽  
Homogeneous Ideal ◽  
Elliptic Singularity ◽  
Rational Surface Singularity ◽  
Generic Matrix ◽  
Image Position ◽  
Coordinate Rings

Let R = k[X1, …, Xn] with k a field, and let I ⊂ R be a homogeneous ideal. The algebra R/I is said to have a pure resolution if its homogeneous minimal resolution has the formSome of the known examples of pure resolutions include the coordinate rings of: the tangent cone of a minimally elliptic singularity or a rational surface singularity [15], a variety defined by generic maximal Pfaffians [2], a variety defined by maximal minors of a generic matrix [3], a variety defined by the submaximal minors of a generic square matrix [6], and certain of the Segre-Veronese varieties [1].If I is in addition Cohen-Macaulay, then Herzog and Kühl have shown that the betti numbers bi are completely determined by the twists di.

Rational normal octavic surfaces with a double line, in space of five dimensions

1933 ◽  
Vol 29 (1) ◽  
pp. 95-102 ◽  
Author(s):  
D. W. Babbage
Keyword(s):  
Rational Surface ◽  
Simple Rule ◽  
Plane Curves ◽  
Double Line ◽  
Five Dimensions ◽  
Multiple Line ◽  
Image Position

The following paper arises from a remark in a recent paper by Professor Baker. In that paper he gives a simple rule, under which a rational surface has a multiple line, expressed in terms of the system of plane curves which represent the prime sections of the surface. The rule is that, if one system of representing curves is given by an equation of the formthe surface being given, in space (x0, x1,…, xr), by the equationsthen the surface contains the linecorresponding to the curve φ = 0; and if the curve φ = 0 has genus q, this line is of multiplicity q + 1.

scholarly journals Pascal type properties of Betti numbers

1994 ◽  
Vol 17 (3) ◽  
pp. 545-552
Author(s):  
Tilak de Alwis
Keyword(s):  
Polynomial Ring ◽  
Betti Numbers ◽  
Binomial Coefficients ◽  
Minimal Resolution ◽  
Pascal’S Triangle ◽  
Specific Formula ◽  
Pascal's Triangle

In this paper, we will describe the Pascal Type properties of Betti numbers of ideals associated ton-gons. These are quite similar to the properties enjoyed by the Pascal's Triangle, concerning the binomial coefficients. By definition, the Betti numbersβt(n)of an idealIassociated to ann-gon are the ranks of the modules in a free minimal resolution of theR-moduleR/I, whereRis the polynomial ringk[x1,x2,…,xn]. Herekis any field andx1,x2,…,xnare indeterminates. We will prove those properties using a specific formula for the Betti numbers.

scholarly journals Frobenius categories, Gorenstein algebras and rational surface singularities

2014 ◽  
Vol 151 (3) ◽  
pp. 502-534 ◽  
Author(s):  
Martin Kalck ◽  
Osamu Iyama ◽  
Michael Wemyss ◽  
Dong Yang
Keyword(s):  
Sufficient Conditions ◽  
Rational Surface ◽  
Gorenstein Ring ◽  
Projective Modules ◽  
Rational Singularity ◽  
Gorenstein Projective Modules ◽  
Surface Singularities ◽  
Frobenius Category ◽  
Rational Surface Singularity ◽  
Frobenius Categories

AbstractWe give sufficient conditions for a Frobenius category to be equivalent to the category of Gorenstein projective modules over an Iwanaga–Gorenstein ring. We then apply this result to the Frobenius category of special Cohen–Macaulay modules over a rational surface singularity, where we show that the associated stable category is triangle equivalent to the singularity category of a certain discrepant partial resolution of the given rational singularity. In particular, this produces uncountably many Iwanaga–Gorenstein rings of finite Gorenstein projective type. We also apply our method to representation theory, obtaining Auslander–Solberg and Kong type results.

Bounds on Betti Numbers

1982 ◽  
Vol 34 (3) ◽  
pp. 589-592
Author(s):  
Mark Ramras
Keyword(s):  
Natural Number ◽  
Local Ring ◽  
Residue Class ◽  
Asymptotic Properties ◽  
Betti Numbers ◽  
Commutative Local Ring ◽  
Artin Ring ◽  
Residue Class Field ◽  
Image Position ◽  
Free Modules

The Betti numbers βn(k) of the residue class field k = R/m of a commutative local ring (R, m) have been studied for about 20 years, primarily as the coefficients of the Poincaré series of E . Several authors have obtained results about the growth of the sequence {βn(k)}.For example, Gulliksen [3] showed that when R is non-regular, the sequence is non-decreasing. More recently, Avramov [1] studied asymptotic properties of {βn(k)} and found that under certain conditions the growth is exponential, i.e., there is a natural number p such that for all n, βpn(k) ≧ 2n.In this paper, we examine the sequence {βn(M)} for arbitrary finitely generated non-free modules M over any commutative local artin ring R. We establish the following bounds:123where l(X) is the length of X.

scholarly journals QUASI-DETERMINANTS AND q-COMMUTING MINORS

2010 ◽  
Vol 52 (3) ◽  
pp. 663-675
Author(s):  
AARON LAUVE
Keyword(s):  
Directed Graph ◽  
Flag Varieties ◽  
Generic Matrix ◽  
Coordinate Rings ◽  
Property Holding

AbstractWe present two new proofs of the q-commuting property holding among certain pairs of quantum minors of a q-generic matrix. The first uses elementary quasi-determinantal arithmetic; the second involves paths in a directed graph. Together, they indicate a means to build the multi-homogeneous coordinate rings of flag varieties in other non-commutative settings.

Classification of Algebraic Surfaces with Sectional Genus less than or Equal to Six. II: Ruled Surfaces with dim ϕKx⊗L(x) = l

1986 ◽  
Vol 38 (5) ◽  
pp. 1110-1121 ◽  
Author(s):  
Elvira Laura Livorni
Keyword(s):  
Line Bundle ◽  
Hyperplane Section ◽  
Ample Line Bundle ◽  
Rational Surface ◽  
Algebraic Surfaces ◽  
Sectional Genus ◽  
Rational Surfaces ◽  
Image Position ◽  
Smooth Hyperplane Section

Let L be a very ample line bundle on a smooth, connected, projective, ruled not rational surface X. We have considered the problem of classifying biholomorphically smooth, connected, projected, ruled, non rational surfaces X with smooth hyperplane section C such that the genus g = g(C) is less than or equal to six and dim where is the map associated to . L. Roth in [10] had given a birational classification of such surfaces. If g = 0 or 1 then X has been classified, see [8].If g = 2 ≠ hl,0(X) by [12, Lemma (2.2.2) ] it follows that X is a rational surface. Thus we can assume g ≦ 3.Since X is ruled, h2,0(X) = 0 andsee [4] and [12, p. 390].

When Is a Complete Ideal in a Rational Surface Singularity a Multiplier Ideal?

2021 ◽  
pp. 145-151
Author(s):  
Maria Alberich-Carramiñana ◽  
Josep Àlvarez Montaner ◽  
Víctor González-Alonso
Keyword(s):  
Rational Surface ◽  
Surface Singularity ◽  
Complete Ideal ◽  
Rational Surface Singularity ◽  
Multiplier Ideal

scholarly journals On symplectic fillings of links of rational surface singularities with reduced fundamental cycle

2004 ◽  
Vol 175 ◽  
pp. 51-57 ◽  
Author(s):  
Mohan Bhupal
Keyword(s):  
Rational Surface ◽  
Surface Singularity ◽  
Fundamental Cycle ◽  
Surface Singularities ◽  
Rational Surface Singularity ◽  
Symplectic Filling ◽  
Rational Surface Singularities

AbstractWe prove that every symplectic filling of the link of a rational surface singularity with reduced fundamental cycle admits a rational compactification, possibly after a modification of the filling in a collar neighbourhood of the link.

On isomorphisms of blowing-ups of complete ideals�of a rational surface singularity

1999 ◽  
Vol 98 (1) ◽  
pp. 65-73 ◽  
Author(s):  
Vincent Cossart ◽  
Olivier Piltant ◽  
Ana J. Reguera-L�pez
Keyword(s):  
Rational Surface ◽  
Surface Singularity ◽  
Rational Surface Singularity
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