Factorial threefold hypersurfaces
Let $X$ be a hypersurface in $\mathbb{P}^{4}$ of degree $d$ that has at most isolated ordinary double points. We prove that $X$ is factorial in the case when $X$ has at most $(d-1)^{2}-1$ singular points.
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arXiv subjects
Ivan Cheltsov
Publications and source records attributed to Ivan Cheltsov.
Nonrational del Pezzo fibrations
Let $X$ be a general divisor in $|3M+nL|$ on the rational scroll $\mathrm{Proj}(\oplus_{i=1}^{4}\mathcal{O}_{\mathbb{P}^{1}}(d_{i}))$, where $d_{i}$ and $n$ are integers, $M$ is the tautological line bundle, $L$ is a fibre of the natural projection to $\mathbb{P}^{1}$, and $d_{1}\geqslant...\geqslant d_{4}=0$. We prove that $X$ is rational $\iff$ $d_{1}=0$ and $n=1$.
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On singular cubic surfaces
We study global log canonical thresholds of cubic surfaces with canonical singularities, and we prove the existence of a Kahler-Einstein metric on two singular cubic surfaces.
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Fano varieties with many selfmaps
We study global log canonical thresholds on anticanonically embedded quasismooth weighted Fano threefold hypersurfaces having terminal quotient singularities to prove the existence of a Kahler-Einstein metric on most of them, and to produce examples of Fano varieties with infinite discrete groups of birational automorphisms.
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Halphen Pencils on Weighted Fano Threefold Hypersurfaces
We classify all pencils on a general weighted hypersurface in $\mathbb{P}(1,a_{1},a_{2},a_{3},a_{4})$ of degree $\sum_{i=1}^{4}a_{i}$ whose general members are surfaces of Kodaira dimension zero.
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Points in projective spaces and applications
We prove the factoriality of a nodal hypersurface in $\mathbb{P}^{4}$ of degree $d$ that has at most $2(d-1)^{2}/3$ singular points, and factoriality of a double cover of $\mathbb{P}^{3}$ branched over a nodal surface of degree $2r$ having less than $(2r-1)r$ singular points.
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Elliptic structures on weighted three-dimensional Fano hypersurfaces
We classify birational maps into elliptic fibrations of a general quasismooth hypersurface in $\mathbb{P}(1,a_{1},a_{2},a_{3},a_{4})$ of degree $\sum_{i=1}^{4}a_{i}$ that has terminal singularities.
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Factorial threefolds and Shokurov vanishing
We prove the factoriality of the following nodal threefolds: a complete intersection of hypersurfaces $F$ and $G\subset\mathbb{P}^{5}$ of degree $n$ and $k$ respectively, where $G$ is smooth, $|\mathrm{Sing}(F\cap G)|\leqslant(n+k-2)(n-1)/5$, $n\geqslant k$; a double cover of a smooth hypersurface $F\subset\mathbb{P}^{4}$ of degree $n$ branched over a surface that is cut out on $F$ by a hypersurface $G$ of degree $2r\geqslant n$, and $|\mathrm{Sing}(F\cap G)|\leqslant(2r+n-2)r/4$.
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On nodal quintic fourfold
We use the Shokurov connectedness principle and the Corti inequality to prove the birational superrigidity of a nodal hypersurface in $\mathbb{P}^{5}$ of degree 5.
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Factorial hypersurfaces in P^4 with nodes
We prove that for n= 5, 6, 7, a nodal hypersurface of degree n in P^4 is factorial if it has at most (n-1)^2-1 nodes.
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On nodal sextic fivefold
We prove the birational superrigidity and nonrationality of a hypersurface in $\mathbb{P}^{6}$ of degree 6 having at most isolated ordinary double points.
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Weighted Fano threefold hypersurfaces
We study birational transformations into elliptic fibrations and birational automorphisms of quasismooth anticanonically embedded weighted Fano 3-fold hypersurfaces having terminal singularities classified by A.R. Iano-Fletcher, J. Johnson, J. Kollar, and M. Reid.
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Double cubics and double quartics
We study a double cover $ψ:X\to V\subset\mathbb{P}^{n}$ branched over a smooth divisor $R\subset V$ such that $R$ is cut on $V$ by a hypersurface of degree $2(n-\mathrm{deg}(V))$, where $n\geqslant 8$ and $V$ is a smooth hypersurface of degree 3 or 4. We prove that $X$ is nonrational and birationally superrigid.
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Hyperelliptic and trigonal Fano threefolds
We classify Fano 3-folds with canonical Gorenstein singularities whose anticanonical linear system has no base points but does not give an embedding, and we classify anticanonically embedded Fano 3-folds with canonical Gorenstein singularities which are not intersections of quadrics. We also study the rationality questions for most of these varieties.
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Sextic Double Solids
We prove non-rationality and birational super-rigidity of a Q-factorial double cover X of P^3 ramified along a sextic surface with at most simple double points. We also show that the condition #|Sing(X)| < 15 implies Q-factoriality of X. In particular, every double cover of P^3 with at most 14 simple double points is non-rational and not birationally isomorphic to a conic bundle. All the birational transformations of X into elliptic fibrations and into Fano 3-folds with canonical singularities are classified. We consider some relevant problems over fields of finite characteristic. When X is defined over a number field F we prove that the set of rational points on the 3-fold X is potentially dense if Sing(X) is not empty.
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Birationally superrigid cyclic triple spaces
We prove the birational superrigidity and the nonrationality of a cyclic triple cover of $\mathbb{P}^{2n}$ branched over a nodal hypersurface of degree $3n$ for $n\ge 2$. In particular, the obtained result solves the problem of the birational superrigidity of smooth cyclic triple spaces. We also consider certain relevant problems.
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On factoriality of nodal threefolds
We prove the $\mathbb{Q}$-factoriality of a nodal hypersurface in $\mathbb{P}^{4}$ of degree $n$ with at most ${\frac{(n-1)^{2}}{4}}$ nodes and the $\mathbb{Q}$-factoriality of a double cover of $\mathbb{P}^{3}$ branched over a nodal surface of degree $2r$ with at most ${\frac{(2r-1)r}{3}}$ nodes.
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Double spaces with isolated singularities
We prove the non-rationality of a double cover of $\mathbb{P}^{n}$ branched over a hypersurface $F\subset\mathbb{P}^{n}$ of degree $2n$ having isolated singularities such that $n\ge 4$ and every singular points of the hypersurface $F$ is ordinary, i.e. the projectivization of its tangent cone is smooth, whose multiplicity does not exceed $2(n-2)$.
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