A Fano $3$-fold with non-rational singularities and a two-dimensional basis

Let (R, m) be a two-dimensional Muhly rational singularity, i.e. the residue field R/m is algebraically closed and the associated graded ring is an integrally closed domain. The goal of this paper is to use immediate quadratic transforms and degree coefficients to investigate complete ideals that are almost adjacent to m, i.e. [Formula: see text].

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ULRICH IDEALS AND MODULES OVER TWO-DIMENSIONAL RATIONAL SINGULARITIES

Nagoya Mathematical Journal ◽

10.1017/nmj.2015.1 ◽

2016 ◽

Vol 221 (1) ◽

pp. 69-110 ◽

Cited By ~ 5

Author(s):

SHIRO GOTO ◽

KAZUHO OZEKI ◽

RYO TAKAHASHI ◽

KEI-ICHI WATANABE ◽

KEN-ICHI YOSHIDA

Keyword(s):

Point Of View ◽

Two Dimensional ◽

Double Points ◽

Rational Singularities ◽

Geometric Point ◽

The Relationship

The main aim of this paper is to classify Ulrich ideals and Ulrich modules over two-dimensional Gorenstein rational singularities (rational double points) from a geometric point of view. To achieve this purpose, we introduce the notion of (weakly) special Cohen–Macaulay modules with respect to ideals, and study the relationship between those modules and Ulrich modules with respect to good ideals.

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Multiplier ideals in two-dimensional local rings with rational singularities

The Michigan Mathematical Journal ◽

10.1307/mmj/1465329014 ◽

2016 ◽

Vol 65 (2) ◽

pp. 287-320 ◽

Cited By ~ 8

Author(s):

Maria Alberich-Carramiñana ◽

Josep Àlvarez Montaner ◽

Ferran Dachs-Cadefau

Keyword(s):

Local Rings ◽

Two Dimensional ◽

Multiplier Ideals ◽

Rational Singularities

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Poincaré series of multiplier ideals in two-dimensional local rings with rational singularities

Advances in Mathematics ◽

10.1016/j.aim.2016.09.008 ◽

2017 ◽

Vol 304 ◽

pp. 769-792 ◽

Cited By ~ 2

Author(s):

Maria Alberich-Carramiñana ◽

Josep Àlvarez Montaner ◽

Ferran Dachs-Cadefau ◽

Víctor González-Alonso

Keyword(s):

Poincaré Series ◽

Local Rings ◽

Two Dimensional ◽

Poincare Series ◽

Multiplier Ideals ◽

Rational Singularities

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CONNECTING RANDOM MATRIX AND TOPOLOGICAL LG MODELS VIA INTEGRABILITY

Modern Physics Letters A ◽

10.1142/s0217732391003304 ◽

1991 ◽

Vol 06 (31) ◽

pp. 2837-2842

Author(s):

K. HERZIG ◽

M. WEISSGOLD ◽

P. TATARU-MIHA

Keyword(s):

Hamiltonian System ◽

Characteristic Polynomial ◽

Double Point ◽

Jacobi Matrix ◽

String Equation ◽

Two Dimensional ◽

Point Singularity ◽

Rational Double Point ◽

Rational Singularities ◽

Double Eigenvalue

We prove that a bi-Hamiltonian system (e.g., a KdV system) is integrable if and only if its Krichever–Moser–Jacobi matrix L has a double eigenvalue, i.e., if (the characteristic polynomial of) L has rational double point singularities. Therefore, L (precisely, its characteristic polynomial) can be identified with the potential of a Landau–Ginzburg (LG) topological model with the same (rational double point) singularity. A rational curve is naturally defined in the corresponding Kummer variety and this explains the appearance (or non-appearance) of the doubling of the string equation as well as a phenomenon observed by Eguchi et al. Finally, we point out a parallelism between rational two-dimensional theories and rational singularities.

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