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A methodology is proposed for nonlinear contrast-enhanced unsupervised segmentation of multispectral (color) microscopy images of principally unstained specimens. The methodology exploits spectral diversity and spatial sparseness to find anatomical differences between materials (cells, nuclei, and background) present in the image. It consists of r th-order rational variety mapping (RVM) followed by matrix/tensor factorization. Sparseness constraint implies duality between nonlinear unsupervised segmentation and multiclass pattern assignment problems. Classes not linearly separable in the original input space become separable with high probability in the higher-dimensional mapped space. Hence, RVM mapping has two advantages: it takes implicitly into account nonlinearities present in the image (ie, they are not required to be known) and it increases spectral diversity (ie, contrast) between materials, due to increased dimensionality of the mapped space. This is expected to improve performance of systems for automated classification and analysis of microscopic histopathological images. The methodology was validated using RVM of the second and third orders of the experimental multispectral microscopy images of unstained sciatic nerve fibers (nervus ischiadicus) and of unstained white pulp in the spleen tissue, compared with a manually defined ground truth labeled by two trained pathophysiologists. The methodology can also be useful for additional contrast enhancement of images of stained specimens.

We show, among other things, that for each integer $$n \ge 3$$ , there is a smooth complex projective rational variety of dimension n, with discrete non-finitely generated automorphism group and with infinitely many mutually non-isomorphic real forms. Our result is inspired by the work of Lesieutre and the work of Dinh and Oguiso.

Let G be an algebraic group over F and p a prime integer. We introduce the notion of a p-retract rational variety and prove that if Y → X is a p-versal G-torsor, then BG is a stable p-retract of X. It follows that the classifying space BG is p-retract rational if and only if there is a p-versal G-torsor Y → X with X a rational variety, that is, all G-torsors over infinite fields are rationally parameterized. In particular, for such groups G the unramified Galois cohomology group $$ {H}_{\mathrm{nr}}^n $$ (F(BG), ℚp/ℤp(j)) coincides with Hn(F, ℚp/ℤp(j)).

В работе исследуются многообразия представлений двух классов конечно порожденных групп.Первый класс состоит из групп с копредставлением\begin{gather*}G = \langle a_1,\ldots,a_s,b_1,\ldots,b_k,x_1,\ldots,x_g\mid\\ a_1^{m_1}=\ldots=a_s^{m_s}= x_1^2\ldots x_g^2 W(a_1,\ldots,a_s,b_1,\ldots,b_k)=1\rangle,\end{gather*}где $g\ge 3$, $m_i\ge 2$ для $i=1,\ldots,s$ и$W(a_1,\ldots,a_s,b_1,\ldots,b_k)$ --- элемент в нормальной формев свободном произведении циклических групп $H=\langle a_1\mid a_1^{m_1}\rangle\ast\ldots\ast\langle a_s\mid a_s^{m_s}\rangle\ast\langle b_1\rangle\ast\ldots\ast\langle b_k\rangle$.Второй класс состоит из групп с копредставлением$$G(p,q) = \langle a_1,\ldots,a_s,b_1,\ldots,b_k,x_1,\ldots,x_g,t\mid a_1^{m_1}=\ldots=a_s^{m_s}=1,\ tU^pt^{-1}=U^q \rangle,$$где $p$ и $q$ --- целые числа, такие, что $p>|q|\geq1$, $(p,q)=1$, $m_i\ge 2$ для $i=1,\ldots,s$, \linebreak $g\ge 3$,$U=x_1^2\ldots x_g^2W(a_1,\ldots,a_s,b_1,\ldots,b_k)$ и $W(a_1,\ldots,a_s,b_1,\ldots,b_k)$ --- элемент, определенный выше.Найдены неприводимые компоненты многообразий представлений $R_n(G)$ и $R_n(G(p,q))$, вычислены их размерности и доказано, что каждая неприводимаякомпонента является рациональным многообразием.

Let $X$ be a smooth Fano threefold. We show that $X$ admits a non-isomorphic surjective endomorphism if and only if $X$ is either a toric variety or a product of $\mathbb{P}^1$ and a del Pezzo surface; in this case, $X$ is a rational variety. We further show that $X$ admits a polarized (or amplified) endomorphism if and only if $X$ is a toric variety., Minor revision, 34 pages, Mathematische Annalen (to appear)

Let$$\mathbb R(C)$$R(C)be the function field of a smooth, irreducible projective curve over$$\mathbb R$$R. LetXbe a smooth, projective, geometrically irreducible variety equipped with a dominant morphismfonto a smooth projective rational variety with a smooth generic fibre over$$\mathbb R(C)$$R(C). Assume that the cohomological obstruction introduced by Colliot-Thélène is the only one to the local-global principle for rational points for the smooth fibres offover$$\mathbb R(C)$$R(C)-valued points. Then we show that the same holds forX, too, by adopting the fibration method similarly to Harpaz–Wittenberg.

By the geometry of the 3-fold quadric we show that the coarse moduli space of genus g ineffective spin hyperelliptic curves with two marked points is a rational variety for every $g \geq 2$., Comment: 29 pages, 3 figures

In a prior paper the authors obtained a four-dimensional discrete integrable dynamical system by the traveling wave reduction from the lattice super-KdV equation in a case of finitely generated Grassmann algebra. The system is a coupling of a Quispel-Roberts-Thompson map and a linear map but does not satisfy the singularity confinement criterion. It was conjectured that the dynamical degree of this system grows quadratically. In this paper, constructing a rational variety where the system is lifted to an algebraically stable map and using the action of the map on the Picard lattice, we prove this conjecture. We also show that invariants can be found through the same technique.

We show that, in any dimension greater than one, there are an abelian variety, a smooth rational variety and a Calabi-Yau manifold, with primitive birational automorphisms of first dynamical degree $>1$. We also show that there are smooth complex projective Calabi-Yau manifolds and smooth rational manifolds, of any even dimension, with primitive biregular automorphisms of positive topological entropy.

We present a new algorithm to decide whether two rational parametric curves are related by a projective transformation and detect all such projective equivalences. Given two rational curves, we derive a system of polynomial equations whose solutions define linear rational transformations of the parameter domain, such that each transformation corresponds to a projective equivalence between the two curves. The corresponding projective mapping is then found by solving a small linear system of equations. Furthermore we investigate the special cases of detecting affine equivalences and symmetries as well as polynomial input curves. The performance of the method is demonstrated by several numerical examples.

Representation and character varieties of the Baumslag–Solitar groups BS(p, q) are analyzed. Irreducible components of these varieties are found, and their dimension is calculated. It is proved that all irreducible components of the representation variety Rn(BS(p, q)) are rational varieties of dimension n2, and each irreducible component of the character variety Xn(BS(p, q)) is a rational variety of dimension k ≤ n. The smoothness of irreducible components of the variety Rns (BS(p, q)) of irreducible representations is established, and it is proved that all irreducible components of the variety Rns (BS(p, q)) are isomorphic to A1 {0}.

We give an explicit description of the divisor class groups of rational trinomial varieties. As an application, we relate the iteration of Cox rings of any rational variety with torus action of complexity one to that of a Du Val surface., Comment: 17 pages

A smooth variety is called uniformly rational if every point admits a Zariski open neighborhood isomorphic to a Zariski open subset of the affine space. In this note we show that every smooth and rational affine variety endowed with an algebraic torus action such that the algebraic quotient has dimension 0 or 1 is uniformly rational., Comment: 4 pages

F. Campana had asked whether a certain threefold is rational. F. Catanese, K. Oguiso and T. T. Truong have recently shown that this variety is birational to a specific conic bundle threefold, which they show is unirational. Computing residues of elements in the Brauer group of the function field of the plane, I prove that that conic bundle threefold is birational to another conic bundle threefold, and the latter is clearly a rational variety.

A linear pentapod is a parallel manipulator with five collinear anchor points on the motion platform (end-effector), which are connected via \(\mathrm {S\underline{P}S}\) legs to the base. This manipulator has five controllable degrees-of-freedom and the remaining one is a free rotation around the motion platform axis (which in fact is an axial spindle). In this paper we present a rational parametrization of the singularity variety of the linear pentapod. Moreover we compute the shortest distance to this rational variety with respect to a suitable metric. Kinematically this distance can be interpreted as the radius of the maximal singularity free-sphere. Moreover we compare the result with the radius of the maximal singularity free-sphere in the position workspace and the orientation workspace, respectively.

Let $M_{g, n}$ (respectively, $\overline{M_{g, n}}$) be the moduli space of smooth (respectively stable) curves of genus $g$ with $n$ marked points. Over the field of complex numbers, it is a classical problem in algebraic geometry to determine whether or not $M_{g, n}$ (or equivalently, $\overline{M_{g, n}}$) is a rational variety. Theorems of J. Harris, D. Mumford, D. Eisenbud and G. Farkas assert that $M_{g, n}$ is not unirational for any $n \geqslant 0$ if $g \geqslant 22$. Moreover, P. Belorousski and A. Logan showed that $M_{g, n}$ is unirational for only finitely many pairs $(g, n)$ with $g \geqslant 1$. Finding the precise range of pairs $(g, n)$, where $M_{g, n}$ is rational, stably rational or unirational, is a problem of ongoing interest. In this paper we address the rationality problem for twisted forms of $\overline{M_{g, n}}$ defined over an arbitrary field $F$ of characteristic $\neq 2$. We show that all $F$-forms of $\overline{M_{g, n}}$ are stably rational for $g = 1$ and $3 \leqslant n \leqslant 4$, $g = 2$ and $2 \leqslant n \leqslant 3$, $g = 3$ and $1 \leqslant n \leqslant 14$, $g = 4$ and $1 \leqslant n \leqslant 9$, $g = 5$ and $1 \leqslant n \leqslant 12$., Comment: 13 pages, proofs much shortened, new coauthor

In this paper we study an instance of projective Reed-Muller type codes, i.e., codes obtained by the evaluation of homogeneous polynomials of a fixed degree in the points of a projective variety. In our case the variety is an important example of a determinantal variety, namely the projective surface known as rational normal scroll, defined over a finite field, which is the basic underlining algebraic structure of this work. We determine the dimension and a lower bound for the minimum distance of the codes, and in many cases we also find the exact value of the minimum distance. To obtain the results we use some methods from Grobner bases theory.

We investigate the relationship between rational plane curves and the envelopes defined by the syzygies of their parameterizations.

Let K be a finite algebraic extension of the field of rational functions in one indeterminate over a finite field and let [Formula: see text] denote an algebraic closure of K. We count points in projective space [Formula: see text] with given height and generating a quadratic extension of K. If n > 2, we derive an asymptotic estimate for the number of such points as the height tends to infinity. Such estimates are analogous to previous results of Schmidt where the field K is replaced by the field of rational numbers ℚ.

Using the Mori theory for threefolds, we prove that the moduli space of pairs of smooth curves of genus four and theta characteristics without global sections is a rational variety.

The investigation of rational varieties with chord length parameterization (shortly RCL varieties) was started by Farin (2006) who observed that rational quadratic circles in standard Bezier form are parametrized by chord length. Motivated by this observation, general RCL curves were studied. Later, the RCL property was extended to rational triangular Bezier surfaces of an arbitrary degree for which the distinguishing property is that the ratios of the three distances of a point to the three vertices of an arbitrary triangle inscribed to the reference circle and the ratios of the distances of the parameter point to the three vertices of the corresponding domain triangle are identical. In this paper, after discussing rational tensor-product surfaces with the RCL property, we present a general unifying approach and study the conditions under which a k-dimensional rational variety in d-dimensional Euclidean space possesses the RCL property. We analyze the entire family of RCL varieties, provide their general parameterization and thoroughly investigate their properties. Finally, the previous observations for curves and surfaces are presented as special cases of the introduced unifying approach.

In this paper we consider the scheme MQ( 2;¡1; 2; 0 ) of stable torsion free sheaves of rank 2 with Chern classes c1 = -1, c2 = 2, c3 = 0 on a smooth 3-dimensional projective quadric Q. The manifold MQ(-1; 2) of moduli bundles of rank 2 with Chern classes c1 = -1, c2 = 2 on Q was studied by Ottaviani and Szurek in 1994. In 2007 the author described the closure MQ (-1; 2) in the scheme MQ(2;¡1; 2; 0). In this paper we prove that in MQ(2;¡1; 2; 0) there exists a unique irreducible component diferent from MQ (¡1; 2) which is a rational variety of dimension 10.

Let \alpha be a Schur root; let h=hcf_v(\alpha(v)) and let p = 1 - < \alpha/h,\alpha/h >. Then a moduli space of representations of dimension vector \alpha is birational to p h by h matrices up to simultaneous conjugacy. Therefore, if h=1,2,3 or 4, then such a moduli space is a rational variety and if h divides 420 it is a stably rational variety.

In this Ph.D. thesis, we investigate some arithmetic properties of algebraic varieties. The thesis consists of two parts: a geometric part (over an arbitrary field) and an arithmetic part (over a number field). The geometric part is devoted to the study of the quotient by its constant part of the third unramified cohomology group of (geometrically) rational surfaces and of their universal torsors. For del Pezzo surfaces of degree at least 5, we show that this quotient is zero, except in the case of del Pezzo surfaces of degree 8 of a special type. For universal torsors as above, we show this quotient is finite and we give a sufficient condition for it to vanish. This condition involves the Galois structure of the geometrical Picard group. The arithmetic part is devoted to the study of the Brauer-Manin obstruction to strong approximation. In collaboration with C. Demarche and F. Xu, we establish the equivalence of étale Brauer-Manin obstruction and the descent obstruction. Then I establish a general theorem about strong approximation of open varieties equipped with an action of a connected linear algebraic group G and containing a G-homogeneous space as open subset.; Dans cette thèse, on s’intéresse à des propriétés arithmétiques des variétés algébriques. Elle contient deux parties : partie géométrique (sur un corps quelconque) et partie arithmétique (sur un corps de nombres). Dans la partie géométrique, j’étudie le quotient par sa partie constante du troisième groupe de cohomologie non ramifiée des surfaces (géométriquement) rationnelles et de leurs torseurs universels. Pour les surfaces de del Pezzo de degré au moins 5, je montre que ce quotient est trivial, sauf pour des surfaces de del Pezzo de degré 8 d’un type particulier. Pour les torseurs universels ci-dessus, je montre que ce quotient est fini et je donne une condition suffisante pour qu’il soit nul, en termes de la structure galoisienne du groupe de Picard géométrique de la surface. Dans la partie arithmétique, on étudie l’obstruction de Brauer–Manin à l’approximation forte. En collaboration avec C. Demarche et F. Xu, nous établissons l’équivalence de l’obstruction de Brauer-Manin étale et de l’obstruction de descente pour les variétés quasi-projectives. Ensuite, j’établis un théorème très général sur l’approximation forte pour les variétés ouvertes munies d’une action d’un groupe linéaire connexe G et dont un ouvert est un espace homogène de G.

In this Ph.D. thesis, we investigate some arithmetic properties of algebraic varieties. The thesis consists of two parts: a geometric part (over an arbitrary field) and an arithmetic part (over a number field). The geometric part is devoted to the study of the quotient by its constant part of the third unramified cohomology group of (geometrically) rational surfaces and of their universal torsors. For del Pezzo surfaces of degree at least 5, we show that this quotient is zero, except in the case of del Pezzo surfaces of degree 8 of a special type. For universal torsors as above, we show this quotient is finite and we give a sufficient condition for it to vanish. This condition involves the Galois structure of the geometrical Picard group. The arithmetic part is devoted to the study of the Brauer-Manin obstruction to strong approximation. In collaboration with C. Demarche and F. Xu, we establish the equivalence of étale Brauer-Manin obstruction and the descent obstruction. Then I establish a general theorem about strong approximation of open varieties equipped with an action of a connected linear algebraic group G and containing a G-homogeneous space as open subset.; Dans cette thèse, on s’intéresse à des propriétés arithmétiques des variétés algébriques. Elle contient deux parties : partie géométrique (sur un corps quelconque) et partie arithmétique (sur un corps de nombres). Dans la partie géométrique, j’étudie le quotient par sa partie constante du troisième groupe de cohomologie non ramifiée des surfaces (géométriquement) rationnelles et de leurs torseurs universels. Pour les surfaces de del Pezzo de degré au moins 5, je montre que ce quotient est trivial, sauf pour des surfaces de del Pezzo de degré 8 d’un type particulier. Pour les torseurs universels ci-dessus, je montre que ce quotient est fini et je donne une condition suffisante pour qu’il soit nul, en termes de la structure galoisienne du groupe de Picard géométrique de la surface. Dans la partie arithmétique, on étudie l’obstruction de Brauer–Manin à l’approximation forte. En collaboration avec C. Demarche et F. Xu, nous établissons l’équivalence de l’obstruction de Brauer-Manin étale et de l’obstruction de descente pour les variétés quasi-projectives. Ensuite, j’établis un théorème très général sur l’approximation forte pour les variétés ouvertes munies d’une action d’un groupe linéaire connexe G et dont un ouvert est un espace homogène de G.

I discuss some arithmetic aspects of higher-dimensional algebraic geometry. I focus on varieties with many rational points and on connections with classification theory and the minimal model program.

Let G be a connected linear algebraic group over an algebraically closed field k, and let H be a connected closed subgroup of G. We prove that the homogeneous variety G/H is a rational variety over k whenever H is solvable, or when dim(G/H) < 11 and characteristic(k) = 0. When H is of maximal rank in G, we also prove that G/H is rational if the maximal semisimple quotient of G is isogenous to a product of almost-simple groups of type A, type C (when characteristic(k) is not 2), or type B_3 or G_2 (when characteristic(k) = 0)., Transactions of the American Mathematical Society (to appear); Lemma 2.2 statement and Lemma 2.6 final part are slightly edited

An irreducible algebraic stack is called \emph{unirational} if there exists a surjective morphism, representable by algebraic spaces, from a rational variety to an open substack. We prove unirationality of the stack of prioritary omalous bundles on Hirzebruch surfaces, which implies also the unirationality of the moduli space of omalous $H$-stable bundles for any ample line bundle $H$ on a Hirzebruch surface. To this end, we find an explicit description of the duals of omalous rank-two bundles with a vanishing condition in terms of monads. Since these bundles are prioritary, we conclude that the stack of prioritary omalous bundles on a Hirzebruch surface different from $\mathbb P^1\times \mathbb P^1$ is dominated by an irreducible section of a Segre variety, and this linear section is rational \cite{I}. In the case of the space quadric, the stack has been explicitly described by N. Buchdahl. As a main tool we use Buchdahl's Beilinson-type spectral sequence. Monad descriptions of omalous bundles on hypersurfaces in $\mathbb P^4$, Calabi-Yau complete intersection, blowups of the projective plane and Segre varieties have been recently obtained by A. A. Henni and M. Jardim~\cite{HJ}, and monads on Hizebruch surfaces have been applied in a different context in~\cite{BBR}., Comment: Final version, to appear in J. Geom. Phys

In the present work we study parameter spaces of two line point configurations introduced by B\"or\"oczky. These configurations are extremal from the point of view of Dirac-Motzkin Conjecture settled recently by Green and Tao. They have appeared also recently in commutative algebra in connection with the containment problem for symbolic and ordinary powers of homogeneous ideals and in algebraic geometry in considerations revolving around the Bounded Negativity Conjecture. Our main results are Theorem A and Theorem B. We show that the parameter space of what we call $B12$ configurations is a three dimensional rational variety. As a consequence we derive the existence of a three dimensional family of rational $B12$ configurations. On the other hand the moduli space of $B15$ configurations is shown to be an elliptic curve with only finitely many rational points, all corresponding to degenerate configurations. Thus, somewhat surprisingly, we conclude that there are no rational $B15$ configurations., Comment: 17 pages, v.2. title modified, material reorganized, introduction new rewritten, discussion more streamlined

We study genus g hyperelliptic curves with reduced automorphism group A 5 and give equations y 2 = f(x) for such curves in both cases where f(x) is a decomposable polynomial in x 2 or x 5. For any fixed genus the locus of such curves is a rational variety. We show that for every point in this locus the field of moduli is a field of definition. Moreover, there exists a rational model y 2 = F(x) or y 2 = x F(x) of the curve over its field of moduli where F(x) can be chosen to be decomposable in x 2 or x 5. While similar equations have been given in (Bujalance et al. in Mm. Soc. Math. Fr. No. 86, 2001) over $${\mathbb R}$$, this is the first time that these equations are given over the field of moduli of the curve.

We prove that in a family of projective threefolds defined over an algebraically closed field, the locus of rational fibers is a countable union of closed subsets of the locus of separably rationally connected fibers. When the ground field has characteristic zero, this implies that the locus of rational fibers in a smooth family of projective threefolds is the union of at most countably many closed subfamilies., Comment: 9 pages; v2: minor changes, final version to appear in Rend. Circ. Mat. Palermo