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Justus Springer

Publications and source records attributed to Justus Springer.

5 recordsLinked to original sources
arXiv2024

Generalizations of Scott's inequality and Pick's formula to rational polygons

We prove a sharp upper bound on the number of boundary lattice points of a rational polygon in terms of its denominator and the number of interior lattice points, generalizing Scott's inequality. We then give sharp lower and upper bounds on the area in terms of the denominator, the number of interior lattice points, and the number of boundary lattice points, which can be seen as a generalization of Pick's formula. Minimizers and maximizers are described in detail. As an application, we derive bounds for the coefficients of Ehrhart quasipolymials of half-integral polygons.

math.CO↗
arXiv2024

Classifying rational polygons with small denominator and few interior lattice points

We present algorithms for classifying rational polygons with fixed denominator and number of interior lattice points. Our approach is to first describe maximal polygons and then compute all subpolygons, where we eliminate redundancy by a suitable normal form. Executing our classification, we obtained a dataset of several billions of polygons covering a wide variety of cases.

math.CO↗
arXiv2023

The Picard index of a surface with torus action

We consider normal rational projective surfaces with torus action and provide a formula for their Picard index, that means the index of the Picard group inside the divisor class group. As an application, we classify the log del Pezzo surfaces with torus action of Picard number one up to Picard index 10,000.

math.AG↗
arXiv2023

Classifying log del Pezzo surfaces with torus action

We consider log del Pezzo surfaces coming with a non-trivial torus action. Such a surface is 1/k-log canonical if it allows a resolution of singularities with discrepanies all greater or equal to 1/k-1. We provide a concrete classification algorithm for fixed k and give explicit results for k=1, 2 and 3. This comprises in particular the cases of Gorenstein index 1, 2 and 3.

math.AG↗