A note on the stability of pencils of plane curves

We investigate the problem of classifying pencils of plane curves of degree d up to projective equivalence. We obtain explicit stability criteria in terms of the log canonical threshold by relating the stability of a pencil to the stability of the curves lying on it.

Theta-regularity and log-canonical threshold

We show that an inequality, proven by Küronya-Pintye, which governs the behavior of the log-canonical threshold of an ideal over $\mathbb {P}^n$ and that of its Castelnuovo-Mumford regularity, can be applied to the setting of principally polarized abelian varieties by substituting the Castelnuovo-Mumford regularity with Θ-regularity of Pareschi-Popa.

HODGE IDEALS FOR -DIVISORS, -FILTRATION, AND MINIMAL EXPONENT

We compute the Hodge ideals of $\mathbb{Q}$ -divisors in terms of the $V$ -filtration induced by a local defining equation, inspired by a result of Saito in the reduced case. We deduce basic properties of Hodge ideals in this generality, and relate them to Bernstein–Sato polynomials. As a consequence of our study we establish general properties of the minimal exponent, a refined version of the log canonical threshold, and bound it in terms of discrepancies on log resolutions, addressing a question of Lichtin and Kollár.

Canonical and log canonical thresholds of multiple projective spaces

We show that the global (log) canonical threshold of d-sheeted covers of the M-dimensional projective space of index 1, where $$d\geqslant 4$$d⩾4, is equal to 1 for almost all families (except for a finite set). The varieties are assumed to have at most quadratic singularities, the rank of which is bounded from below, and to satisfy the regularity conditions. This implies birational rigidity of new large classes of Fano–Mori fibre spaces over a base, the dimension of which is bounded from above by a constant that depends (quadratically) on the dimension of the fibre only.

Lowest log canonical thresholds of a reduced plane curve of degree d

We describe the sixth worst singularity that a plane curve of degree $$d\geqslant 5$$ d ⩾ 5 could have, using its log canonical threshold at the point of singularity. This is an extension of a result due to Cheltsov (J Geom Anal 27(3):2302–2338, 2017) wherein the five lowest values of log canonical thresholds of a plane curve of degree $$d \geqslant 3$$ d ⩾ 3 were computed. These six small log canonical thresholds, in order, are 2 / d, $$({2d-3})/{(d-1)^2}$$ ( 2 d - 3 ) / ( d - 1 ) 2 , $$({2d-1})/(d^2-d)$$ ( 2 d - 1 ) / ( d 2 - d ) , $$({2d-5})/({d^2-3d+1})$$ ( 2 d - 5 ) / ( d 2 - 3 d + 1 ) , $$({2d-3})/(d^2-2d)$$ ( 2 d - 3 ) / ( d 2 - 2 d ) and $$({2d-7})/({d^2-4d+1})$$ ( 2 d - 7 ) / ( d 2 - 4 d + 1 ) . We give examples of curves with these values as their log canonical thresholds using illustrations.

JET SCHEMES OF QUASI-ORDINARY SURFACE SINGULARITIES

We describe the irreducible components of the jet schemes with origin in the singular locus of a two-dimensional quasi-ordinary hypersurface singularity. A weighted graph is associated with these components and with their embedding dimensions and their codimensions in the jet schemes of the ambient space. We prove that the data of this weighted graph is equivalent to the data of the topological type of the singularity. We also determine a component of the jet schemes (equivalent to a divisorial valuation on $\mathbb{A}^{3}$ ), that computes the log-canonical threshold of the singularity embedded in $\mathbb{A}^{3}$ . This provides us with pairs $X\subset \mathbb{A}^{3}$ whose log-canonical thresholds are not computed by monomial divisorial valuations. Note that for a pair $C\subset \mathbb{A}^{2}$ , where $C$ is a plane curve, the log-canonical threshold is always computed by a monomial divisorial valuation (in suitable coordinates of $\mathbb{A}^{2}$ ).

Learning Coefficient of Vandermonde Matrix-Type Singularities in Model Selection

In recent years, selecting appropriate learning models has become more important with the increased need to analyze learning systems, and many model selection methods have been developed. The learning coefficient in Bayesian estimation, which serves to measure the learning efficiency in singular learning models, has an important role in several information criteria. The learning coefficient in regular models is known as the dimension of the parameter space over two, while that in singular models is smaller and varies in learning models. The learning coefficient is known mathematically as the log canonical threshold. In this paper, we provide a new rational blowing-up method for obtaining these coefficients. In the application to Vandermonde matrix-type singularities, we show the efficiency of such methods.