Rebecca Lehman | The Hebrew University of Jerusalem (original) (raw)

Papers by Rebecca Lehman

Research paper thumbnail of Algebraic invariants in classification of 6-points in degenerations of surfaces

In this paper, we find isomorphisms between certain invariant groups corresponding to different n... more In this paper, we find isomorphisms between certain invariant groups corresponding to different numerations on 6-points of surfaces. There is a combinatorial correspondence between four 6-point orderings obtained by exchanging two opposite labels. We derive isomorphisms between certain invariant quotient groups obtained from these 6-point nu-merations. This is a preliminary step towards an ultimate classification of 6-points invariants, and perhaps towards a proof that the invariant groups, or at least certain derived invariants, are independent of the arbitrary choice of the numeration.

Research paper thumbnail of Fax (812)-877-8333 Phone (812)-877-8193 Hyperbolic Billiard Paths

A useful way to investigate closed geodesics on a kaleidoscopically tiled surface is to look at t... more A useful way to investigate closed geodesics on a kaleidoscopically tiled surface is to look at the billiard path described by a closed geodesic on a single tile. When looking at billiard paths it is possible to ignore surfaces and restrict ourselves to the tiling of the hyperbolic plane. We classify the smallest billiard paths by wordlength and parity. We also demonstrate the existence of orientable paths and investigate conjectures about the billiard

Research paper thumbnail of Hyperbolic Billiard Paths

A useful way to investigate closed geodesics on a kaleidoscopically tiled surface is to look at t... more A useful way to investigate closed geodesics on a kaleidoscopically tiled surface is to look at the billiard path described by a closed geodesic on a single tile. When looking at billiard paths it is possible to ignore surfaces and restrict ourselves to the tiling of the hyperbolic plane. We classify the smallest billiard paths by wordlength and parity. We also demonstrate the existence of orientable paths and investigate conjectures about the billiard spectrum for the (2, 3, 7) tiling.

Research paper thumbnail of Brill-Noether-type Theorems with a Movable Ramification Point

arXiv: Algebraic Geometry, 2007

The classical Brill-Noether theorems count the dimension of the family of maps from a general cur... more The classical Brill-Noether theorems count the dimension of the family of maps from a general curve of genus g to non-degenerate curves of degree d in r-dimensional projective space. These theorems can be extended to include ramification conditions at fixed general points. This paper deals with the problem of imposing a ramification condition at an unspecified point. We solve the problem completely in dimensions 1 and 2, and provide an existence test and bound the dimension of the family in the general case.

Research paper thumbnail of Algebraic invariants in classiflcation of 6-points in degenerations of surfaces

In this paper, we flnd isomorphisms between certain invari- ant groups corresponding to difierent... more In this paper, we flnd isomorphisms between certain invari- ant groups corresponding to difierent numerations on 6-points of surfaces. There is a combinatorial correspondence between four 6-point orderings obtained by exchanging two opposite labels. We derive isomorphisms be- tween certain invariant quotient groups obtained from these 6-point nu- merations. This is a preliminary step towards an ultimate classiflcation of 6-points invariants, and perhaps towards a proof that the invariant groups, or at least certain derived invariants, are independent of the arbi- trary choice of the numeration.

Research paper thumbnail of Finding Alphas on Futures and Forwards

Research paper thumbnail of Risk and Drawdowns

Research paper thumbnail of Classification of fundamental groups of Galois covers of surfaces of small degree

Let X be a surface of degree n, projected onto CP 2. The surface has a natural Galois cover with ... more Let X be a surface of degree n, projected onto CP 2. The surface has a natural Galois cover with Galois group S n. It is possible to determine the fundamental group of a Galois cover from that of the complement of the branch curve of X. In this paper we survey the fundamental groups of Galois covers of all surfaces of small degree n ≤ 4, that degenerate to a nice plane arrangement, namely a union of n planes such that no three planes meet in a line. We include the already classical examples of the quadric, the Hirzebruch and the Veronese surfaces and the degree 4 embedding of CP 1 ×CP 1 , and also add new computations for the remaining cases: the cubic embedding of the Hirzebruch surface F 1 , the Cayley cubic (or a smooth surface in the same family), for a quartic surface that degenerates to the union of a triple point and a plane not through the triple point, and for a quartic 4-point. In an appendix, we also include the degree 8 surface CP 1 ×CP 1 embedded by the (2, 2) embedding, and the degree 2n surface embedded by the (1, n) embedding, in order to complete the classification of all embeddings of CP 1 × CP 1 , which was begun in [23]. Partially supported by the Emmy Noether Research Institute for Mathematics (center of the Minerva Foundation of Germany), the Excellency Center "Group Theoretic Methods in the Study of Algebraic Varieties" of the Israel Science Foundation, and EAGER (EU network, HPRN-CT-2009-00099).

Research paper thumbnail of Brill-Noether-type Theorems with a Movable Ramification Point

Eprint Arxiv 0804 4657, Apr 1, 2008

The classical Brill-Noether theorems count the dimension of the family of maps from a general cur... more The classical Brill-Noether theorems count the dimension of the family of maps from a general curve of genus g to non-degenerate curves of degree d in r-dimensional projective space. These theorems can be extended to include ramification conditions at fixed general points. This paper deals with the problem of imposing a ramification condition at an unspecified point. We solve the problem completely in dimensions 1 and 2, and provide an existence test and bound the dimension of the family in the general case.

Research paper thumbnail of Classification of fundamental groups of Galois covers of surfaces of small degree degenerating to nice plane arrangements

Contemporary Mathematics, 2011

Let X be a surface of degree n, projected onto CP 2. The surface has a natural Galois cover with ... more Let X be a surface of degree n, projected onto CP 2. The surface has a natural Galois cover with Galois group S n. It is possible to determine the fundamental group of a Galois cover from that of the complement of the branch curve of X. In this paper we survey the fundamental groups of Galois covers of all surfaces of small degree n ≤ 4, that degenerate to a nice plane arrangement, namely a union of n planes such that no three planes meet in a line. We include the already classical examples of the quadric, the Hirzebruch and the Veronese surfaces and the degree 4 embedding of CP 1 ×CP 1 , and also add new computations for the remaining cases: the cubic embedding of the Hirzebruch surface F 1 , the Cayley cubic (or a smooth surface in the same family), for a quartic surface that degenerates to the union of a triple point and a plane not through the triple point, and for a quartic 4-point. In an appendix, we also include the degree 8 surface CP 1 ×CP 1 embedded by the (2, 2) embedding, and the degree 2n surface embedded by the (1, n) embedding, in order to complete the classification of all embeddings of CP 1 × CP 1 , which was begun in [23]. Partially supported by the Emmy Noether Research Institute for Mathematics (center of the Minerva Foundation of Germany), the Excellency Center "Group Theoretic Methods in the Study of Algebraic Varieties" of the Israel Science Foundation, and EAGER (EU network, HPRN-CT-2009-00099).

Research paper thumbnail of On ramified covers of the projective plane II: Generalizing Segre’s theory

Journal of the European Mathematical Society, 2000

The classical Segre theory gives a necessary and sufficient condition for a plane curve to be a b... more The classical Segre theory gives a necessary and sufficient condition for a plane curve to be a branch curve of a (generic) projection of a smooth surface in P 3. We generalize this result for smooth surfaces in a projective space of any dimension in the following way: given two plane curves, B and E, we give a necessary and sufficient condition for B to be the branch curve of a surface X in P N and E to be the image of the double curve of a P 3-model of X. In the classical Segre theory, a plane curve B is a branch curve of a smooth surface in P 3 iff its 0-cycle of singularities is special with respect to a linear system of plane curves of particular degree. Here we prove that B is a branch curve of a surface in P N iff (part of) the cycle of singularities of the union of B and E is special with respect to the linear system of plane curves of a particular low degree. In particular, given just a curve B, we provide some necessary conditions for B to be a branch curve of a smooth surface in P N .

Research paper thumbnail of Classification of Fundamental Groups of Galois Covers of Surfaces of Small Degree Degenerating to Nice Plane Arrangements

Arxiv preprint arXiv: …, 2010

Let X be a surface of degree n, projected onto CP 2. The surface has a natural Galois cover with ... more Let X be a surface of degree n, projected onto CP 2. The surface has a natural Galois cover with Galois group S n. It is possible to determine the fundamental group of a Galois cover from that of the complement of the branch curve of X. In this paper we survey the fundamental groups of Galois covers of all surfaces of small degree n ≤ 4, that degenerate to a nice plane arrangement, namely a union of n planes such that no three planes meet in a line. We include the already classical examples of the quadric, the Hirzebruch and the Veronese surfaces and the degree 4 embedding of CP 1 ×CP 1 , and also add new computations for the remaining cases: the cubic embedding of the Hirzebruch surface F 1 , the Cayley cubic (or a smooth surface in the same family), for a quartic surface that degenerates to the union of a triple point and a plane not through the triple point, and for a quartic 4-point. In an appendix, we also include the degree 8 surface CP 1 ×CP 1 embedded by the (2, 2) embedding, and the degree 2n surface embedded by the (1, n) embedding, in order to complete the classification of all embeddings of CP 1 × CP 1 , which was begun in [23]. Partially supported by the Emmy Noether Research Institute for Mathematics (center of the Minerva Foundation of Germany), the Excellency Center "Group Theoretic Methods in the Study of Algebraic Varieties" of the Israel Science Foundation, and EAGER (EU network, HPRN-CT-2009-00099).

Research paper thumbnail of Algebraic invariants in classification of 6-points in degenerations of surfaces

In this paper, we find isomorphisms between certain invariant groups corresponding to different n... more In this paper, we find isomorphisms between certain invariant groups corresponding to different numerations on 6-points of surfaces. There is a combinatorial correspondence between four 6-point orderings obtained by exchanging two opposite labels. We derive isomorphisms between certain invariant quotient groups obtained from these 6-point nu-merations. This is a preliminary step towards an ultimate classification of 6-points invariants, and perhaps towards a proof that the invariant groups, or at least certain derived invariants, are independent of the arbitrary choice of the numeration.

Research paper thumbnail of Fax (812)-877-8333 Phone (812)-877-8193 Hyperbolic Billiard Paths

A useful way to investigate closed geodesics on a kaleidoscopically tiled surface is to look at t... more A useful way to investigate closed geodesics on a kaleidoscopically tiled surface is to look at the billiard path described by a closed geodesic on a single tile. When looking at billiard paths it is possible to ignore surfaces and restrict ourselves to the tiling of the hyperbolic plane. We classify the smallest billiard paths by wordlength and parity. We also demonstrate the existence of orientable paths and investigate conjectures about the billiard

Research paper thumbnail of Hyperbolic Billiard Paths

A useful way to investigate closed geodesics on a kaleidoscopically tiled surface is to look at t... more A useful way to investigate closed geodesics on a kaleidoscopically tiled surface is to look at the billiard path described by a closed geodesic on a single tile. When looking at billiard paths it is possible to ignore surfaces and restrict ourselves to the tiling of the hyperbolic plane. We classify the smallest billiard paths by wordlength and parity. We also demonstrate the existence of orientable paths and investigate conjectures about the billiard spectrum for the (2, 3, 7) tiling.

Research paper thumbnail of Brill-Noether-type Theorems with a Movable Ramification Point

arXiv: Algebraic Geometry, 2007

The classical Brill-Noether theorems count the dimension of the family of maps from a general cur... more The classical Brill-Noether theorems count the dimension of the family of maps from a general curve of genus g to non-degenerate curves of degree d in r-dimensional projective space. These theorems can be extended to include ramification conditions at fixed general points. This paper deals with the problem of imposing a ramification condition at an unspecified point. We solve the problem completely in dimensions 1 and 2, and provide an existence test and bound the dimension of the family in the general case.

Research paper thumbnail of Algebraic invariants in classiflcation of 6-points in degenerations of surfaces

In this paper, we flnd isomorphisms between certain invari- ant groups corresponding to difierent... more In this paper, we flnd isomorphisms between certain invari- ant groups corresponding to difierent numerations on 6-points of surfaces. There is a combinatorial correspondence between four 6-point orderings obtained by exchanging two opposite labels. We derive isomorphisms be- tween certain invariant quotient groups obtained from these 6-point nu- merations. This is a preliminary step towards an ultimate classiflcation of 6-points invariants, and perhaps towards a proof that the invariant groups, or at least certain derived invariants, are independent of the arbi- trary choice of the numeration.

Research paper thumbnail of Finding Alphas on Futures and Forwards

Research paper thumbnail of Risk and Drawdowns

Research paper thumbnail of Classification of fundamental groups of Galois covers of surfaces of small degree

Let X be a surface of degree n, projected onto CP 2. The surface has a natural Galois cover with ... more Let X be a surface of degree n, projected onto CP 2. The surface has a natural Galois cover with Galois group S n. It is possible to determine the fundamental group of a Galois cover from that of the complement of the branch curve of X. In this paper we survey the fundamental groups of Galois covers of all surfaces of small degree n ≤ 4, that degenerate to a nice plane arrangement, namely a union of n planes such that no three planes meet in a line. We include the already classical examples of the quadric, the Hirzebruch and the Veronese surfaces and the degree 4 embedding of CP 1 ×CP 1 , and also add new computations for the remaining cases: the cubic embedding of the Hirzebruch surface F 1 , the Cayley cubic (or a smooth surface in the same family), for a quartic surface that degenerates to the union of a triple point and a plane not through the triple point, and for a quartic 4-point. In an appendix, we also include the degree 8 surface CP 1 ×CP 1 embedded by the (2, 2) embedding, and the degree 2n surface embedded by the (1, n) embedding, in order to complete the classification of all embeddings of CP 1 × CP 1 , which was begun in [23]. Partially supported by the Emmy Noether Research Institute for Mathematics (center of the Minerva Foundation of Germany), the Excellency Center "Group Theoretic Methods in the Study of Algebraic Varieties" of the Israel Science Foundation, and EAGER (EU network, HPRN-CT-2009-00099).

Research paper thumbnail of Brill-Noether-type Theorems with a Movable Ramification Point

Eprint Arxiv 0804 4657, Apr 1, 2008

The classical Brill-Noether theorems count the dimension of the family of maps from a general cur... more The classical Brill-Noether theorems count the dimension of the family of maps from a general curve of genus g to non-degenerate curves of degree d in r-dimensional projective space. These theorems can be extended to include ramification conditions at fixed general points. This paper deals with the problem of imposing a ramification condition at an unspecified point. We solve the problem completely in dimensions 1 and 2, and provide an existence test and bound the dimension of the family in the general case.

Research paper thumbnail of Classification of fundamental groups of Galois covers of surfaces of small degree degenerating to nice plane arrangements

Contemporary Mathematics, 2011

Let X be a surface of degree n, projected onto CP 2. The surface has a natural Galois cover with ... more Let X be a surface of degree n, projected onto CP 2. The surface has a natural Galois cover with Galois group S n. It is possible to determine the fundamental group of a Galois cover from that of the complement of the branch curve of X. In this paper we survey the fundamental groups of Galois covers of all surfaces of small degree n ≤ 4, that degenerate to a nice plane arrangement, namely a union of n planes such that no three planes meet in a line. We include the already classical examples of the quadric, the Hirzebruch and the Veronese surfaces and the degree 4 embedding of CP 1 ×CP 1 , and also add new computations for the remaining cases: the cubic embedding of the Hirzebruch surface F 1 , the Cayley cubic (or a smooth surface in the same family), for a quartic surface that degenerates to the union of a triple point and a plane not through the triple point, and for a quartic 4-point. In an appendix, we also include the degree 8 surface CP 1 ×CP 1 embedded by the (2, 2) embedding, and the degree 2n surface embedded by the (1, n) embedding, in order to complete the classification of all embeddings of CP 1 × CP 1 , which was begun in [23]. Partially supported by the Emmy Noether Research Institute for Mathematics (center of the Minerva Foundation of Germany), the Excellency Center "Group Theoretic Methods in the Study of Algebraic Varieties" of the Israel Science Foundation, and EAGER (EU network, HPRN-CT-2009-00099).

Research paper thumbnail of On ramified covers of the projective plane II: Generalizing Segre’s theory

Journal of the European Mathematical Society, 2000

The classical Segre theory gives a necessary and sufficient condition for a plane curve to be a b... more The classical Segre theory gives a necessary and sufficient condition for a plane curve to be a branch curve of a (generic) projection of a smooth surface in P 3. We generalize this result for smooth surfaces in a projective space of any dimension in the following way: given two plane curves, B and E, we give a necessary and sufficient condition for B to be the branch curve of a surface X in P N and E to be the image of the double curve of a P 3-model of X. In the classical Segre theory, a plane curve B is a branch curve of a smooth surface in P 3 iff its 0-cycle of singularities is special with respect to a linear system of plane curves of particular degree. Here we prove that B is a branch curve of a surface in P N iff (part of) the cycle of singularities of the union of B and E is special with respect to the linear system of plane curves of a particular low degree. In particular, given just a curve B, we provide some necessary conditions for B to be a branch curve of a smooth surface in P N .

Research paper thumbnail of Classification of Fundamental Groups of Galois Covers of Surfaces of Small Degree Degenerating to Nice Plane Arrangements

Arxiv preprint arXiv: …, 2010

Let X be a surface of degree n, projected onto CP 2. The surface has a natural Galois cover with ... more Let X be a surface of degree n, projected onto CP 2. The surface has a natural Galois cover with Galois group S n. It is possible to determine the fundamental group of a Galois cover from that of the complement of the branch curve of X. In this paper we survey the fundamental groups of Galois covers of all surfaces of small degree n ≤ 4, that degenerate to a nice plane arrangement, namely a union of n planes such that no three planes meet in a line. We include the already classical examples of the quadric, the Hirzebruch and the Veronese surfaces and the degree 4 embedding of CP 1 ×CP 1 , and also add new computations for the remaining cases: the cubic embedding of the Hirzebruch surface F 1 , the Cayley cubic (or a smooth surface in the same family), for a quartic surface that degenerates to the union of a triple point and a plane not through the triple point, and for a quartic 4-point. In an appendix, we also include the degree 8 surface CP 1 ×CP 1 embedded by the (2, 2) embedding, and the degree 2n surface embedded by the (1, n) embedding, in order to complete the classification of all embeddings of CP 1 × CP 1 , which was begun in [23]. Partially supported by the Emmy Noether Research Institute for Mathematics (center of the Minerva Foundation of Germany), the Excellency Center "Group Theoretic Methods in the Study of Algebraic Varieties" of the Israel Science Foundation, and EAGER (EU network, HPRN-CT-2009-00099).