Geometric genus

Geometric genus is like counting the number of holes in a shape, but it's a special kind of shape called an algebraic curve. Imagine you have a circle, it has no holes, so it has a genus of 0. But if you take a donut shape, it has one hole, so it has a genus of 1.

Now, let's move on to a more complicated shape like an algebraic curve. It's kind of like a twisted loop-de-loop made from a string that's been tied in different ways. This shape might look like it has holes, but it's not always that easy to tell. The geometric genus helps us figure out if this shape really has holes or not.

To calculate the geometric genus, we use a special formula that involves something called the degree and the arithmetic genus. The degree is like the size of the loop-de-loop, and the arithmetic genus is like a score we give the shape based on its complexity.

Once we have these numbers, we use the formula to figure out the geometric genus. If the number is 0, then the shape doesn't have any holes, but if the number is higher, then the shape has one or more holes.

So, in simpler terms, just think of geometric genus as a way to count the number of holes in a really complicated shape, like a twisted string loop-de-loop. It uses a special formula that involves the size and complexity of the shape to determine if it has holes or not.