Existence of Curves with Prescribed Topological Singularities

Thomas Keilen; Ilya Tyomkin

Home/Archive/arXiv

arXiv · math/0104062

Existence of Curves with Prescribed Topological Singularities

Thomas Keilen·Ilya Tyomkin

View original source ↗Read source document ↗Export metadata ↓

Abstract

Throughout this paper we study the existence of irreducible curves C on smooth projective surfaces S with singular points of prescribed topological types S_1,...,S_r. There are necessary conditions for the existence of the type \sum_{i=1}^r μ(S_i) < aC^2+bC.K+c+1 for some fixed divisor K on S and suitable coefficients a, b and c, and the main sufficient condition that we find is of the same type, saying it is asymptotically optimal. Even for the case where S is the projective plane, ten years ago general results of this quality have not been known. An important ingredient for the proof is a vanishing theorem for invertible sheaves on the blown up S of the form O_{S'}(π^*D-\sum_{i=1}^r m_iE_i), deduced from the Kawamata-Vieweg Vanishing Theorem. Its proof covers the first part of the paper, while the middle part is devoted to the existence theorems. In the last part we investigate our conditions on ruled surfaces, products of elliptic curves, surfaces in projective 3-space, and K3-surfaces.

Explore related subjects

math.AG

Keep this discovery

Explore connections, maps & timelines

BibTeX RIS

Thomas Keilen, Ilya Tyomkin. 2001-12-07. Existence of Curves with Prescribed Topological Singularities. https://arxiv.org/abs/math/0104062

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Braid and Phantom

Let N be the moduli space of stable rank 2 vector bundles on a smooth projective curve of genus g>1 with fixed odd determinant. With Sebastian Torres, we previously found a semi-orthogonal decomposition of the bounded derived category of N into bounded derived categories of symmetric powers of the curve and, possibly, a phantom block. In this work, we employ the theory of weaving patterns to eliminate the possibility of a phantom, completing the proof of the decomposition conjectured by Narasimhan and, independently, by Belmans, Galkin, and Mukhopadhyay.

math.AG↗

arXiv2026

On the Alexander polynomials of conic-line arrangements

In the present paper we compute Alexander polynomials for certain classes of conic-line arrangements in the complex projective plane which are related to pencils. We prove two general results for curve arrangements coming from Halphen pencils of index $k\geq 2$. Then we apply them to the Hesse arrangement of conics and to some of its degenerations. The results are completed by computations using computer algebra. In particular, we construct conic-line arrangements which are non-reduced pencil-type arrangements and have as roots of their Alexander polynomials roots of unity of order 7. Such roots are not known and are conjectured not to exist in the class of line arrangements.

math.AG↗

arXiv2026

On the Tensor Property of Bernstein-Sato Polynomial

We prove the multiplicative Thom-Sebastiani rule for Bernstein-Sato polynomials, answering the longstanding questions of Budur and Popa. We generalize the result to the tensor of two effective divisors on the product of two arbitrary non-singular complex varieties. This also leads to a multiplicative property related to Igusa's strong monodromy conjecture. Moreover, we propose an extension of our result to Bernstein-Sato polynomials for ideals and prove it for monomial ideals.

math.AG↗