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Genuinely ramified maps and monodromy

Indranil Biswas, Manish Kumar, A. J. Parameswaran

Abstract

For any genuinely ramified morphism $f\, :\, Y\, \longrightarrow\, X$ between irreducible smooth projective curves we prove that $\overline{(Y\times_X Y) \setminus Δ}$ is connected, where $Δ\, \subset\, Y\times_X Y$ is the diagonal. Using this result the following are proved: If $f$ is further Morse then the Galois closure is the symmetric group $S_d$, where $d\,=\, \text{degree}(f)$. The Galois group of the general projection, to a line, of any smooth curve $X\,\subset\, \PP^n$ of degree $d$, which is not contained in a hyperplane and contains a non-flex point, is $S_d$.

Genuinely ramified maps and monodromy

Abstract

For any genuinely ramified morphism between irreducible smooth projective curves we prove that is connected, where is the diagonal. Using this result the following are proved: If is further Morse then the Galois closure is the symmetric group , where . The Galois group of the general projection, to a line, of any smooth curve of degree , which is not contained in a hyperplane and contains a non-flex point, is .
Paper Structure (3 sections, 7 theorems, 23 equations)

This paper contains 3 sections, 7 theorems, 23 equations.

Table of Contents

  1. Introduction
  2. Morse map
  3. Genuinely ramified morphisms

Key Result

Lemma 2.1

Let $f\,:\,Y\,\longrightarrow\, X$ be a Morse map. Then $Y'$ in g0 is smooth. If $Y'$ is also connected, then the restriction $q'_1\, :=\, q_1\vert_{Y'} \, :\, Y'\,\longrightarrow\, Y$ of $q_1$ (see g1) is again a Morse map.

Theorems & Definitions (16)

  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Theorem 3.1
  • proof
  • Corollary 3.2
  • proof
  • Corollary 3.3
  • proof
  • ...and 6 more