A. Tikhomirov

activity

19982025

most citedThe Abel-Jacobi map of a moduli component of vector bundles on the cubic threefold

83 citations · 180 across the 24 of their papers we have counts for

collaborators

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25 papers · 1 filter

math.AG2024

Unirationality of instanton moduli space for small charges

Dimitri Markushevich, Alexander Tikhomirov

The unirationality of the moduli space of mathematical instantons on the projective 3-space is proved for charges less than or equal to 7.

math.AG2024

Moduli of rank two semistable sheaves on rational Fano threefolds of the main series

Alexander S. Tikhomirov, Danil A. Vassiliev

In this paper we investigate the moduli spaces of semistable coherent sheaves of rank two on the projective space $P^{3}$ and the following rational Fano manifolds of the ma…

math.AG2019★ 1 cited

An algebraic-geometric construction of ind-varieties of generalized flags

A. Tikhomirov, I. Penkov

We define the class of admissible linear embeddings of flag varieties. The definition is given in the general language of algebraic geometry. We then prove that an admissible linea…

math.AG2019★ 10 cited

Irreducible components of the moduli space of rank 2 sheaves of odd determinant on the projective space

Charles Almeida, Marcos Jardim, Alexander S. Tikhomirov

We describe new irreducible components of the moduli space of rank $2$ semistable torsion free sheaves on the three-dimensional projective space whose generic point corresponds to…

math.AG2018★ 1 cited

Construction of stable rank 2 vector bundles on $P^{3}$ via symplectic bundles

Alexander Tikhomirov, Sergey Tikhomirov, Danil Vasiliev

In this article we study the Gieseker-Maruyama moduli spaces $B (e, n)$ of stable rank 2 algebraic vector bundles with Chern classes $c_{1} = e \in {- 1, 0}, c_{2} = n \geq 1$ on the p…

math.AG2017★ 6 cited

Semistable rank 2 sheaves with singularities of mixed dimension on $P^{3}$

Alexey N. Ivanov, Alexander S. Tikhomirov

We describe new irreducible components of the Gieseker-Maruyama moduli scheme $M (3)$ of semistable rank 2 coherent sheaves with Chern classes $c_{1} = 0, c_{2} = 3, c_{3} = 0$ on $…

math.AG2017

Series of rational moduli components of stable rank 2 vector bundles on $P^{3}$

Alexey Kytmanov, Alexander Tikhomirov, Sergey Tikhomirov

We study the problem of rationality of an infinite series of components, the so-called Ein components, of the Gieseker-Maruyama moduli space $M (e, n)$ of rank 2 stable vector bundle…

math.AG2017★ 15 cited

Moduli spaces of rank 2 instanton sheaves on the projective space

Marcos Jardim, Mario Maican, Alexander S. Tikhomirov

We study the irreducible components of the moduli space of instanton sheaves on $P^{3}$ , that is rank 2 torsion free sheaves $E$ with $c_{1} (E) = c_{3} (E) = 0$ satisfying $h^1(E(-2)…

math.AG2017★ 9 cited

New moduli components of rank 2 bundles on projective space

Charles Almeida, Marcos Jardim, Alexander Tikhomirov +1

We present a new family of monads whose cohomology is a stable rank two vector bundle on $P^{3}$ . We also study the irreducibility and smoothness together with a geometrical…

math.AG2016★ 23 cited

Two infinite series of moduli spaces of rank 2 sheaves on $P^{3}$

Marcos Jardim, Dimitri Markushevich, Alexander S. Tikhomirov

We describe new components of the Gieseker--Maruyama moduli scheme $M (n)$ of semistable rank 2 sheaves $E$ on $P^{3}$ with $c_{1} (E) = 0$ , $c_{2} (E) = n$ and $c_{3} (E) = 0$ wh…

math.AG2015

New divisors in the boundary of the instanton moduli space

Marcos Jardim, Dimitri Markushevich, Alexander S. Tikhomirov

Let $I (n)$ denote the moduli space of rank $2$ instanton bundles of charge $n$ on $P^{3}$ . We know from several authors that $I (n)$ is an irreducible,…

math.AG2014★ 1 cited

Symplectic instanton bundles on P3 and 't Hooft instantons

Ugo Bruzzo, Dimitri Markushevich, Alexander Tikhomirov

We study the moduli space $I_{n, r}$ of rank- $2 r$ symplectic instanton vector bundles on $P^{3}$ with $r \geq 2$ and second Chern class $n \geq r + 1, n - r \equiv 1 (mod 2)$ .…

math.AG2014★ 5 cited

On the Barth-Van de Ven-Tyurin-Sato theorem

Ivan Penkov, Alexander S. Tikhomirov

The Barth-Van de Ven-Tyurin-Sato Theorem claims that any finite rank vector bundle on the infinite complex projective space $P^{\infty}$ is isomorphic to a direct sum of lin…

math.AG2013

Linear ind-Grassmannians

Ivan Penkov, Alexander S. Tikhomirov

We consider ind-varieties obtained as direct limits of chains of embeddings $X_1\stackrel{ϕ_1}{\hookrightarrow}\dots\stackrel{ϕ_{m-1}}{\hookrightarrow} X_m\stackrel{ϕ_m}{\hookright…

math.AG2011

Bubble tree compactification of moduli spaces of vector bundles on surfaces

D. Markushevich, A. S. Tikhomirov, G. Trautmann

In this article we announce some results on compactifying moduli spaces of rank-2 vector bundles on surfaces by spaces of vector bundles on trees of surfaces. This is thought as an…

math.AG2011★ 1 cited

Moduli of symplectic instanton vector bundles of higher rank on projective space $P^{3}$

Ugo Bruzzo, D. Markushevich, A. S. Tikhomirov

Symplectic instanton vector bundles on the projective space $P^{3}$ constitute a natural generalization of mathematical instantons of rank 2. We study the moduli space $I_{n…

math.AG2011★ 5 cited

Algebraic cycles on quadric sections of cubics in P^4 under the action of symplectomorphisms

V. Guletskii, A. Tikhomirov

We consider the involution changing the sign of two coordinates in 4-dimensional projective space. The intersection S of invariant cubic and quadric hypersurfaces in P^4 is a K3-su…

math.AG2010★ 7 cited

Rationality of instanton moduli

D. Markushevich, A. S. Tikhomirov

Tikhomirov (2009) proved the irreducibility of the moduli space of mathematical instantons on the projective 3-space for all odd charges. The irreducibility for charges between 1 a…

math.AG2010★ 2 cited

Uhlenbeck-Donaldson compactification for framed sheaves on projective surfaces

Ugo Bruzzo, Dimitri Markushevich, Alexander Tikhomirov

We construct a compactification $M^{μ ss}$ of the Uhlenbeck-Donaldson type for the moduli space of slope stable framed bundles. This is a kind of a moduli space of slope semistable…

math.AG2007

Rank 2 vector bundles on ind-Grassmannians

Ivan Penkov, Alexander S. Tikhomirov

The simplest example of an ind-Grassmannian is the infinite projective space $P^{\infty}$ . The Barth-Van de Ven-Tyurin (BVT) Theorem, proved more than 30 years ago \cite{BV},…

math.AG2007★ 1 cited

Triviality of vector bundles on sufficiently twisted ind-Grassmannians

Ivan Penkov, Alexander S. Tikhomirov

Twisted ind-Grassmannians are ind-varieties $\GG$ obtained as direct limits of Grassmannians $G (r_{m}, V^{r_{m}})$ , for $m\in\ZZ_{>0}$, under embeddings $ϕ_m:G(r_m,V^{r_m})\to G(r_{m+1}…

math.AG2002★ 2 cited

A parametrization of the theta divisor of the quartic double solid

D. Markushevich, A. S. Tikhomirov

Let M(2;0,3) be the moduli space of rank-2 stable vector bundles with Chern classes c_1=0, c_2=3 on the Fano threefold X, the double solid of index two. We prove that the vector bu…

math.AG2002★ 4 cited

Irreducibility and Smoothness of the moduli space of mathematical 5--instantons over $P_{3}$

I. Coanda, A. Tikhomirov, G. Trautmann

We prove that the space of mathematical instantons with second Chern class 5 over $P_{3}$ is smooth and irreducible. Unified and simple proofs for the same statements in ca…

math.AG1999

Symplectic structure on a moduli space of sheaves on the cubic fourfold

D. Markushevich, A. S. Tikhomirov

A 10-dimensional symplectic moduli space of torsion sheaves on the cubic 4-fold is constructed. It parametrizes the stable rank 2 vector bundles on the hypeplane sections of the cu…

Unirationality of instanton moduli space for small charges

Moduli of rank two semistable sheaves on rational Fano threefolds of the main series

An algebraic-geometric construction of ind-varieties of generalized flags

Irreducible components of the moduli space of rank 2 sheaves of odd determinant on the projective space

Construction of stable rank 2 vector bundles on P3 via symplectic bundles

Semistable rank 2 sheaves with singularities of mixed dimension on P3

Series of rational moduli components of stable rank 2 vector bundles on P3

Moduli spaces of rank 2 instanton sheaves on the projective space

New moduli components of rank 2 bundles on projective space

Two infinite series of moduli spaces of rank 2 sheaves on P3

New divisors in the boundary of the instanton moduli space

Symplectic instanton bundles on P3 and 't Hooft instantons

On the Barth-Van de Ven-Tyurin-Sato theorem

Linear ind-Grassmannians

Bubble tree compactification of moduli spaces of vector bundles on surfaces

Moduli of symplectic instanton vector bundles of higher rank on projective space P3

Algebraic cycles on quadric sections of cubics in P^4 under the action of symplectomorphisms

Rationality of instanton moduli

Uhlenbeck-Donaldson compactification for framed sheaves on projective surfaces

Rank 2 vector bundles on ind-Grassmannians

Triviality of vector bundles on sufficiently twisted ind-Grassmannians

A parametrization of the theta divisor of the quartic double solid

Irreducibility and Smoothness of the moduli space of mathematical 5--instantons over P3​

Symplectic structure on a moduli space of sheaves on the cubic fourfold

Construction of stable rank 2 vector bundles on $P^{3}$ via symplectic bundles

Semistable rank 2 sheaves with singularities of mixed dimension on $P^{3}$

Series of rational moduli components of stable rank 2 vector bundles on $P^{3}$

Two infinite series of moduli spaces of rank 2 sheaves on $P^{3}$

Moduli of symplectic instanton vector bundles of higher rank on projective space $P^{3}$

Irreducibility and Smoothness of the moduli space of mathematical 5--instantons over $P_{3}$