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Given n general points p_1, p_2, \\ldots , p_n \\in \\mathbb P^r, it is natural to ask when there exists a curve C \\subset \\mathbb P^r, of degree d and genus g, passing through p_1, p_2, \\ldots , p_n. In this paper, the authors give a complete answer to this question for curves C with nonspecial hyperplane section. This result is a consequence of our main theorem, which states that the normal bundle N_C of a general nonspecial curve of degree d and genus g in \\mathbb P^r (with d \\geq g + r) has the property of interpolation (i.e. that for a general effective divisor D of any degree on C, either H^0(N_C(-D)) = 0 or H^1(N_C(-D)) = 0), with exactly three exceptions.

Comprehensive descriptions of animal behavior require precise three-dimensional (3D) measurements of whole-body movements. Although two-dimensional approaches can track visible landmarks in restrictive environments, performance drops in freely moving animals, due to occlusions and appearance changes. Therefore, we designed DANNCE to robustly track anatomical landmarks in 3D across species and behaviors. DANNCE uses projective geometry to construct inputs to a convolutional neural network that leverages learned 3D geometric reasoning. We trained and benchmarked DANNCE using a dataset of nearly seven million frames that relates color videos and rodent 3D poses. In rats and mice, DANNCE robustly tracked dozens of landmarks on the head, trunk, and limbs of freely moving animals in naturalistic settings. We extended DANNCE to datasets from rat pups, marmosets, and chickadees, and demonstrate quantitative profiling of behavioral lineage during development.DANNCE enables robust 3D tracking of animals’ limbs and other features in naturalistic environments by making use of a deep learning approach that incorporates geometric reasoning. DANNCE is demonstrated on behavioral sequences from rodents, marmosets, and chickadees.

In this paper, we study (i)-curves with i∈−1,0,1 in the blown-up projective space P[sup.r] in general points. The notion of (−1)-curves was analyzed in the early days of mirror symmetry by Kontsevich, with the motivation of counting curves on a Calabi–Yau threefold. In dimension two, Nagata studied planar (−1)-curves in order to construct a counterexample to Hilbert’s 14th problem. We introduce the notion of classes of (0)- and (1)-curves in P[sup.r] with s points blown up, and we prove that their number is finite if and only if the space is a Mori Dream Space. We further introduce a bilinear form on a space of curves and a unique symmetric Weyl-invariant class, F (which we will refer to as the anticanonical curve class ). For Mori Dream Spaces, we prove that (−1)-curves can be defined arithmetically by the linear and quadratic invariants determined by the bilinear form. Moreover, (0)- and (1)-Weyl lines give the extremal rays for the cone of movable curves in P[sup.r] with r+3 points blown up. As an application, we use the technique of movable curves to reprove that if F[sup.2] ≤0 then Y is not a Mori Dream Space, and we propose to apply this technique to other spaces.

Anosov representations of word hyperbolic groups into higher-rank semisimple Lie groups are representations with finite kernel and discrete image that have strong analogies with convex cocompact representations into rank-one Lie groups. However, the most naive analogy fails: generically, Anosov representations do not act properly and cocompactly on a convex set in the associated Riemannian symmetric space. We study representations into projective indefinite orthogonal groups PO(p,q) by considering their action on the associated pseudo-Riemannian hyperbolic space Hp,q-1 in place of the Riemannian symmetric space. Following work of Barbot and Mérigot in anti-de Sitter geometry, we find an intimate connection between Anosov representations and a natural notion of convex cocompactness in this setting.

Recursive constructions derived from finite projective geometries generate scalable families of resolvable block designs exhibiting strong algebraic regularity and intrinsic symmetry. In this work, we investigate the structural optimization of recursion depth in such constructions and demonstrate that the combinatorial growth of recursive chains is governed by a quadratic scaling law originating from the asymptotic expansion of Gaussian binomial coefficients. We show that the resulting exponent is strictly concave, which guarantees the existence and uniqueness of an optimal recursion depth. This optimum occurs at the midpoint of the projective dimension and reflects the dual symmetry of the lattice of projective subspaces. To analyze this behavior, we introduce a normalized objective function that compares recursion depths and reveals a unique maximum corresponding to the midpoint of the projective dimension. Theoretical analysis is supported by exact enumeration and asymptotic validation, confirming that the optimal depth is robust to lower-order perturbations and remains invariant under the natural duality of projective geometry. The proposed framework establishes a direct connection between symmetry properties of discrete geometric structures and optimality in nonlinear discrete systems. These results provide a unified perspective on recursive design constructions, revealing that symmetry not only governs combinatorial structure but also induces a mathematically inevitable optimal configuration. The approach opens new directions for studying symmetry-induced optimality in combinatorial geometry, discrete optimization, and related nonlinear mathematical models.

It is proved that, for any vector space V over a field f of finite dimension at least 3, the projective space P(V) (the set of all subspaces of V equpped with a binary predicate of inclusion) is regularly injectively bi-interpretable with the field F.

The authors provide a complete classification of globally generated vector bundles with first Chern class $c_1 \\leq 5$ one the projective plane and with $c_1 \\leq 4$ on the projective $n$-space for $n \\geq 3$. This reproves and extends, in a systematic manner, previous results obtained for $c_1 \\leq 2$ by Sierra and Ugaglia [J. Pure Appl. Algebra 213 (2009), 2141-2146], and for $c_1 = 3$ by Anghel and Manolache [Math. Nachr. 286 (2013), 1407-1423] and, independently, by Sierra and Ugaglia [J. Pure Appl. Algebra 218 (2014), 174-180]. It turns out that the case $c_1 = 4$ is much more involved than the previous cases, especially on the projective 3-space. Among the bundles appearing in our classification one can find the Sasakura rank 3 vector bundle on the projective 4-space (conveniently twisted). The authors also propose a conjecture concerning the classification of globally generated vector bundles with $c_1 \\leq n - 1$ on the projective $n$-space. They verify the conjecture for $n \\leq 5$.

This volume contains the proceedings of the Workshop on Motivic Homotopy Theory and Refined Enumerative Geometry, held from May 14-18, 2018, at the Universität Duisburg-Essen, Essen, Germany. It constitutes an accessible yet swift introduction to a new and active area within algebraic geometry, which connects well with classical intersection theory. Combining both lecture notes aimed at the graduate student level and research articles pointing towards the manifold promising applications of this refined approach, it broadly covers refined enumerative algebraic geometry.

It is well known that pseudo–Riemannian metrics in the projective class of a given torsion free affine connection can be obtained from (and are equivalent to) the solutions of a certain overdetermined projectively invariant differential equation. This equation is a special case of a so-called first Bernstein–Gelfand–Gelfand (BGG) equation. The general theory of such equations singles out a subclass of so-called normal solutions. We prove that non-degenerate normal solutions are equivalent to pseudo–Riemannian Einstein metrics in the projective class and observe that this connects to natural projective extensions of the Einstein condition.

We construct in projective differential geometry of the real dimension 2 higher symmetry algebra of the symplectic Dirac operator acting on symplectic spinors. The higher symmetry differential operators correspond to the solution space of a class of projectively invariant overdetermined operators of arbitrarily high order acting on symmetric tensors. The higher symmetry algebra structure corresponds to a completely prime primitive ideal having as its associated variety the minimal nilpotent orbit of sl(3,R) .