Ilya Tyomkin

Multi-linear Secret-Sharing Schemes

Multi-linear secret-sharing schemes are the most common secret-sharing schemes. In these schemes the secret is composed of some field elements and the sharing is done by applying some fixed linear mapping on the field elements of the secret and some randomly chosen field elements. If the secret contains one field element, then the scheme is called linear. The importance of multi-linear schemes is that they provide a simple non-interactive mechanism for computing shares of linear combinations of previously shared secrets. Thus, they can be easily used in cryptographic protocols.

Existence of curves with prescribed topological singularities

Throughout this paper we study the existence of irreducible curves C on smooth projective surfaces Σ with singular points of prescribed topological types S 1 ,…,S r . There are necessary conditions for the existence of the type Σ r i=1 μ(S i ) < αC 2 +βC.K+γ for some fixed divisor K on Σ and suitable coefficients α, β and γ, and the main sufficient condition that we find is of the same type, saying it is asymptotically proper. Ten years ago general results of this quality were not known even for the case E = P 2 C An important ingredient for the proof is a vanishing theorem for invertible sheaves on the blown up E of the form O Σ (π * D-Σ r i=1 m i E i ), deduced from the Kawamata-Vieweg Vanishing Theorem. Its proof covers the first part of the paper, while the middle part is devoted to the existence theorems. In the last part we investigate our conditions on ruled surfaces, products of elliptic curves, surfaces in P 3 C , and K3-surfaces.

On Zariski's theorem in positive characteristic

In the current paper we show that the dimension of a family

Ferrand pushouts for algebraic spaces

V

Enumeration of rational curves with cross-ratio constraints

of irreducible reduced curves in a given ample linear system on a toric surface

Tropical geometry and correspondence theorems via toric stacks

S

On the quantum homology algebra of toric Fano manifolds

over an algebraically closed field is bounded from above by

On Severi Varieties on Hirzebruch Surfaces

-K_S.C+p_g(C)-1

Versal deformation of algebraic hypersurfaces with isolated singularities

, where

Prüfer algebraic spaces

C