Related papers: Lower semi-continuity of the Waldschmidt constants

We prove invariant of the regularity index of fat points under changes of the linear subspace containing the support of the fat points. Then we show that Segre's bound is attained by any set of s non-degenerate equimultiple fat points in…

In this paper we study semi-stable, radially symmetric and decreasing solutions $u\in W^{1,p}(B_1)$ of $-\Delta_p u=g(u)$ in $B_1\setminus\{0\}$, where $B_1$ is the unit ball of $\mathbb{R}^N$, $p>1$, $\Delta_p$ is the $p-$Laplace operator…

Motivated by questions in interpolation theory and on linear systems of rational varieties, one is interested in upper bounds for the Castelnuovo-Mumford regularity of arbitrary subschemes of fat points. An optimal upper bound, named after…

This paper surveys certain problems involving numerical characters for ideals I(Z) defining fat points subschemes $Z=m_1p_1+...+m_np_n$ for general points $p_i\in {\bf P}^2$. It also presents some new results, and includes a suite of…

We demonstrate a measure theoretical approach to the local regularity of weak supersolutions to elliptic and parabolic equations in divergence form. In the first part, we show that weak supersolutions become lower semicontinuous after…

Let Z be a set of fat points in P^1 x P^1 that is also arithmetically Cohen-Macaulay (ACM). We describe how to compute the degree of a separator of a fat point of multiplicity m for each point in the support of Z using only a numerical…

In this article we prove the following theorems about weak approximation of smooth cubic hypersurfaces and del Pezzo surfaces of degree 4 defined over global fields. (1) For cubic hypersurfaces defined over global function fields, if there…

In this note we address the relation between symbolic and ordinary powers of the ideal of a reduced set or points in projective space: the so-called containment problem. In particular, we obtain sharp lower bounds on the Waldschmidt…

For a symplectic manifold $M$ let $\{\cdot,\cdot\}$ be the corresponding Poisson bracket. In this note we prove that the functional $(F,G) \mapsto \|\{F,G\}\|_{L^p(M)}$ is lower-semicontinuous with respect to the $C^0$-norm on…

Let I = p_1^{m_1} \cap ... \cap p_s^{m_s} be the defining ideal of a scheme of fat points in P^{n_1} x ... x P^{n_k} with support in generic position. When all the m_i's are 1, we explicitly calculate the Castelnuovo-Mumford regularity of…

In the present note we study Waldschmidt constants of Stanley-Reisner ideals of a hypergraph and a graph with vertices forming a bipyramid over a planar n-gon. The case of the hypergraph has been studied by Bocci and Franci. We reprove…

We study del Pezzo surfaces of degree 1 of the form w^2 = z^3 + Ax^6 + By^6 in the weighted projective space P_k(1,1,2,3), where k is a perfect field of characteristic not 2 or 3 and A,B \in k^*. Over a number field, we exhibit an infinite…

We describe the GIT compactification of the moduli of (2,2)-type effective divisors of $\mathbb{P}^1\times\mathbb{P}^2$ (i.e., surfaces of the linear system $\vert \pi_1^*\mathcal{O}_{\mathbb{P}^1}(2)\otimes…

We establish the enumerativity of (original and modified) Welschinger invariants for every real divisor on any real algebraic Del Pezzo surface and give an algebro-geometric proof of the invariance of that count both up to variation of the…

We consider a class of abstract quasilinear parabolic problems with lower--order terms exhibiting a prescribed singular structure. We prove well--posedness and Lipschitz continuity of associated semiflows. Moreover, we investigate global…

For a scheme of fat points $Z$ defined by the saturated ideal $\mathcal{I}_Z$, the regularity index computes the Castelnuovo-Mumford regularity of the Cohen-Macaulay ring $R/\mathcal{I}_Z.$ For points in " general position" we improve the…

In this note we improve a result of Steffens on the lower bound for Seshadri constants in very general points of a surface with one-dimensional N\'eron-Severi space. We also show a multi-point counterpart of such a lower bound.

We obtain Lipschitz estimates for bounded minimizers of functionals with nonstandard $(p,q)$-growth satisfying the dimension-independent restriction $q<p+2$ with $p \geq 2$. This relation improves existing restrictions when $p \leq N-1$,…

In this paper we are concerned with a sequence of univariate random variables with piecewise polynomial means and independent sub-Gaussian noise. The underlying polynomials are allowed to be of arbitrary but fixed degrees. All the other…

We prove that the integral points are potentially Zariski dense in the complement of a reduced effective singular anticanonical divisor in a smooth del Pezzo surface, with the exception of $\mathbb{P}^2$ minus three concurrent lines (for…