I am supposed to do a presentation on Lefschetz hyperplane section theorem via Morse theory (following Milnor's Morse Theory) for my algebraic geometry class...I more or less understand the proof, but I am really at a loss what could be good and easily presentable applications of the theorem. I am a beginner in algebraic geometry, so I can't do anything highly involved. Griffiths and Harris has a couple of examples....I was wondering if you people could suggest something really interesting and elegant... thanks a lot.

Dear anonymous, let $ X \subset \mathbb P^n(\mathbb C)$ be a smooth hypersurface of degree $d$ in projective space. Let me show how Lefschetz directly calculates the homology $H_i(X)=H_i(X,\mathbb Z)$ of $X$, except for $H_{n1}(X)$. There is an embedding $v:\mathbb P^n(\mathbb C) \to \mathbb P^N(\mathbb C)$, called the Veronese embedding, mapping the original $P^n(\mathbb C)$ to a huge projective space $\mathbb P^N(\mathbb C)$ of dimension $N=\binom{n+d}{d} 1$ You thus get $H_i(X) \simeq H_i(P)$ for $i\leq n2$ by applying Lefschetz for homology. For $i\geq n$ you deduce (by applying Poincaré duality and Lefschetz for cohomology ) $H_i(X) \simeq H^{2n2i}(X)\simeq H^{2n2i}(\mathbb P^n(\mathbb C))$. Since $P\simeq P^n(\mathbb C)$ all the above homology groups are easy to calculate. Indeed, for $0 \leq i \leq 2n$, projective space has homology $H_i(P^n(\mathbb C))$ alternately $0$ and $\mathbb Z$. Ditto for cohomology. The only group which has partly escaped us is $H_{n1}(X)$, for which Lefschetz only tells us that it surjects onto $H_{n1}(P^n(\mathbb C))$. As a reality check, notice that for $n=2$ we know that the rank of $H_{1}(X)$ is $2g=(d1)(d2)$, where $g$ is the genus of our curve $X$ . This is indeed not equal to the rank of any (co)homology group of $P^2(\mathbb C)$ for $d \geq 3$ Optional complement: Allow me to show how concrete the Veronese embedding is. It maps the point
$(x_0:x_1:x_2:\cdots:x_n)\in \mathbb P^n$ to the point $(\cdots:M_\delta (x):\cdots) \in \mathbb P^N$ where $\delta = (d_0,\cdots,d_n) \quad d_0+\cdots d_n=d$ and $M_\delta (x)=x_0^{d_0}\cdots x_n^{d_n}$. 


Here is an important application. It is probably in the relevant chapter of Voisins book. Let $V^k \subset CP^n$ be a smooth projective variety, $d$ a number and $s$ be a section of the $d$th power of the Hopf bundle on $CP^n$. Assume that $s$ is transverse to the zero section and that $V$ and $s^{1}(0)$ are transverse. Let $W$ be the intersection. Then $\pi_i(W) \to \pi_i(V)$ is an iso if $i< k$. If $d$ were $1$, then this is just the hyperplane theorem. For larger $d$, the sections of $H^{\otimes d}$ define an embedding $f$ into a higherdimensional projective space $P^{N(d)}$ (the Veronese embedding, see Georges answer) and the section $s$ defines a hyperplane in that larger space. Then apply the hyperpalane theorem to $f(W) \to f(V)$; and you get the result. A complete intersection variety $V^k \subset P^n$ is the common zero set of $nk$ sections $s_1, \ldots,s_{nk}$; $s_i$ a section in $H^{\otimes d_i}$ and these sections are in general position. By an iterative application you get that $\pi_i (V)=\pi_i(P^n)$ for $i< k$. Hence projective tori of dimension $>1$ are not complete intersections. 

