For each integer a = 2,3,4, we construct a family of minimal complex algebraic surfaces with torsion group Z/2 such that c1 = 2χ(O)−1 and χ(O) = a, where c1 and χ(O) are the first Chern class and the Euler characteristic of the structure sheaf.

In this paper, I construct noncompact analogs of the Chern classes of equivariant vector bundles over complex reductive groups. For the tangent bundle, these Chern classes yield an adjunction formula for the Euler characteristic of complete intersections in reductive groups. In the case where the complete intersection is a curve, this formula gives an explicit answer for the Euler characteristi...

Let V be a closed subscheme of a projective space P. We give an algorithm to compute the Chern-Schwartz-MacPherson class, and the Euler characteristic of V and an algorithm to compute the Segre class of V . The algorithms can be implemented using either symbolic or numerical methods. The algorithms are based on a new method for calculating the projective degrees of a rational map defined by a h...

1 f-vector for convex polytopes Given a convex polytope P of full dimension in R, we can consider the f-vector, fk = number of k-faces, 0 ≤ k ≤ d− 1. A natural question is Question 1. Given an abstract tuple f of d integers, what are necessary and sufficient conditions for f to be the f-vector of a convex polytope? It turns out to be quite difficult to give sufficient conditions, so let’s deter...

where V , E, and F are, respectively, the number of vertices, edges, and faces of p. The formula was first stated in print by Euler in 1758 [11]. The proof given here is based on Poincaré’s linear algebraic proof, stated in [17] (with a corrected proof in [18]), as adapted by Imre Lakatos in the latter’s Proofs and Refutations [15]. As is well known, Euler’s formula is not true for all polyhedr...

We show that the maximum likelihood degree of a smooth very affine variety is equal to the signed topological Euler characteristic. This generalizes Orlik and Terao’s solution to Varchenko’s conjecture on complements of hyperplane arrangements to smooth very affine varieties. For very affine varieties satisfying a genericity condition at infinity, the result is further strengthened to relate th...

We find a new class of data for which the Prandtl boundary layer equations and the hydrostatic Euler equations are locally in time well-posed. In the case of the Prandtl equations, we assume that the initial datum u0 is monotone on a number of intervals (on some strictly increasing on some strictly decreasing) and analytic on the complement and show that the local existence and uniqueness hold....

We prove a criterion of nonsingularity of a complete intersection of two fiberwise quadrics in P P 1 (O(d 1) ⊕. .. ⊕ O(d 5)). As a corollary we derive the following addition to the Alexeev theorem on rationality of standard Del Pezzo fibrations of degree 4 over P 1 : we prove that any fibration of this kind with the topological Euler characteristic χ(X) = −4 is rational.

In this note, we give an explicit counterexample to the simple loop conjecture for representations of surface groups into PSL(2, R). Specifically, we use a construction of DeBlois and Kent to show that for any orientable surface with negative Euler characteristic and genus at least 1, there are uncountably many non-conjugate, non-injective homomorphisms of its fundamental group into PSL(2, R) t...

It is known that if G is a graph that can be drawn without edges crossing in a surface with Euler characteristic ǫ, and k and d are positive integers such that k > 3 and d is sufficiently large in terms of k and ǫ, then G is (k, d)∗-colorable; that is, the vertices of G can be colored with k colors so that each vertex has at most d neighbors with the same color as itself. In this paper, the kno...